### Skills

- Marko currently has 20 tulips in his yard. Each year he plants 5 more.
- Write a recursive formula for the number of tulips Marko has
- Write an explicit formula for the number of tulips Marko has

- Pam is a Disc Jockey. Every week she buys 3 new albums to keep her collection current. She currently owns 450 albums.
- Write a recursive formula for the number of albums Pam has
- Write an explicit formula for the number of albums Pam has

- A store’s sales (in thousands of dollars) grow according to the recursive rule Pn=Pn-1 + 15, with initial population P0=40.
- Calculate P1 and P2
- Find an explicit formula for Pn
- Use your formula to predict the store’s sales in 10 years
- When will the store’s sales exceed $100,000?

- The number of houses in a town has been growing according to the recursive rule Pn=Pn-1 + 30, with initial population P0=200.
- Calculate P1 and P2
- Find an explicit formula for Pn
- Use your formula to predict the number of houses in 10 years
- When will the number of houses reach 400 houses?

- A population of beetles is growing according to a linear growth model. The initial population (week 0) was P0=3, and the population after 8 weeks is P8=67.
- Find an explicit formula for the beetle population in week n
- After how many weeks will the beetle population reach 187?

- The number of streetlights in a town is growing linearly. Four months ago (
*n*= 0) there were 130 lights. Now (*n*= 4) there are 146 lights. If this trend continues,- Find an explicit formula for the number of lights in month
*n* - How many months will it take to reach 200 lights?

- Find an explicit formula for the number of lights in month
- Tacoma’s population in 2000 was about 200 thousand, and had been growing by about 9% each year.
- Write a recursive formula for the population of Tacoma
- Write an explicit formula for the population of Tacoma
- If this trend continues, what will Tacoma’s population be in 2016?
- When does this model predict Tacoma’s population to exceed 400 thousand?

- Portland’s population in 2007 was about 568 thousand, and had been growing by about 1.1% each year.
- Write a recursive formula for the population of Portland
- Write an explicit formula for the population of Portland
- If this trend continues, what will Portland’s population be in 2016?
- If this trend continues, when will Portland’s population reach 700 thousand?

- Diseases tend to spread according to the exponential growth model. In the early days of AIDS, the growth rate was around 190%. In 1983, about 1700 people in the U.S. died of AIDS. If the trend had continued unchecked, how many people would have died from AIDS in 2005?
- The population of the world in 1987 was 5 billion and the annual growth rate was estimated at 2 percent per year. Assuming that the world population follows an exponential growth model, find the projected world population in 2015.
- A bacteria culture is started with 300 bacteria. After 4 hours, the population has grown to 500 bacteria. If the population grows exponentially,
- Write a recursive formula for the number of bacteria
- Write an explicit formula for the number of bacteria
- If this trend continues, how many bacteria will there be in 1 day?
- How long does it take for the culture to triple in size?

- A native wolf species has been reintroduced into a national forest. Originally 200 wolves were transplanted. After 3 years, the population had grown to 270 wolves. If the population grows exponentially,
- Write a recursive formula for the number of wolves
- Write an explicit formula for the number of wolves
- If this trend continues, how many wolves will there be in 10 years?
- If this trend continues, how long will it take the population to grow to 1000 wolves?

- One hundred trout are seeded into a lake. Absent constraint, their population will grow by 70% a year. The lake can sustain a maximum of 2000 trout. Using the logistic growth model,
- Write a recursive formula for the number of trout
- Calculate the number of trout after 1 year and after 2 years.

- Ten blackberry plants started growing in my yard. Absent constraint, blackberries will spread by 200% a month. My yard can only sustain about 50 plants. Using the logistic growth model,
- Write a recursive formula for the number of blackberry plants in my yard
- Calculate the number of plants after 1, 2, and 3 months

- In 1968, the U.S. minimum wage was $1.60 per hour. In 1976, the minimum wage was $2.30 per hour. Assume the minimum wage grows according to an exponential model where
*n*represents the time in years after 1960.- Find an explicit formula for the minimum wage.
- What does the model predict for the minimum wage in 1960?
- If the minimum wage was $5.15 in 1996, is this above, below or equal to what the model predicts?

### Concepts

- The population of a small town can be described by the equation
*Pn*= 4000 + 70*n*, where*n*is the number of years after 2005. Explain in words what this equation tells us about how the population is changing. - The population of a small town can be described by the equation
*Pn*= 4000(1.04)*n*, where*n*is the number of years after 2005. Explain in words what this equation tells us about how the population is changing.

### Exploration

Most of the examples in the text examined growing quantities, but linear and exponential equations can also describe decreasing quantities, as the next few problems will explore.

- A new truck costs $32,000. The car’s value will depreciate over time, which means it will lose value. For tax purposes, depreciation is usually calculated linearly. If the truck is worth $24,500 after three years, write an explicit formula for the value of the car after
*n*years. - Inflation causes things to cost more, and for our money to buy less (hence your grandparents saying, “In my day, you could buy a cup of coffee for a nickel”). Suppose inflation decreases the value of money by 5% each year. In other words, if you have $1 this year, next year it will only buy you $0.95 worth of stuff. How much will $100 buy you in 20 years?
- Suppose that you have a bowl of 500 M&M candies, and each day you eat ¼ of the candies you have. Is the number of candies left changing linearly or exponentially? Write an equation to model the number of candies left after
*n*days. - A warm object in a cooler room will decrease in temperature exponentially, approaching the room temperature according to the formula where
*Tn*is the temperature after*n*minutes,*r*is the rate at which temperature is changing,*a*is a constant, and*Tr*is the temperature of the room. Forensic investigators can use this to predict the time of death of a homicide victim. Suppose that when a body was discovered (*n =*0) it was 85 degrees. After 20 minutes, the temperature was measured again to be 80 degrees. The body was in a 70 degree room.- Use the given information with the formula provided to find a formula for the temperature of the body.
- When did the victim die, if the body started at 98.6 degrees?

- Recursive equations can be very handy for modeling complicated situations for which explicit equations would be hard to interpret. As an example, consider a lake in which 2000 fish currently reside. The fish population grows by 10% each year, but every year 100 fish are harvested from the lake by people fishing.
- Write a recursive equation for the number of fish in the lake after
*n*years. - Calculate the population after 1 and 2 years. Does the population appear to be increasing or decreasing?
- What is the maximum number of fish that could be harvested each year without causing the fish population to decrease in the long run?

- Write a recursive equation for the number of fish in the lake after
- The number of Starbucks stores grew after first opened. The number of stores from 1990-2007, as reported on their corporate website
^{[1]}, is shown below.- Carefully plot the data. Does is appear to be changing linearly or exponentially?
- Try finding an equation to model the data by picking two points to work from. How well does the equation model the data?
- Try using an equation of the form , where
*k*is a constant, to model the data. This type of model is called a Power model. Compare your results to the results from part b.*Note: to use this model, you will need to have 1990 correspond with n = 1 rather than n = 0.*

Year | Number of Starbucks stores | Year | Number of Starbucks stores |

1990 | 84 | 1999 | 2498 |

1991 | 116 | 2000 | 3501 |

1992 | 165 | 2001 | 4709 |

1993 | 272 | 2002 | 5886 |

1994 | 425 | 2003 | 7225 |

1995 | 677 | 2004 | 8569 |

1996 | 1015 | 2005 | 10241 |

1997 | 1412 | 2006 | 12440 |

1998 | 1886 | 2007 | 15756 |

- Thomas Malthus was an economist who put forth the principle that population grows based on an exponential growth model, while food and resources grow based on a linear growth model. Based on this, Malthus predicted that eventually demand for food and resources would out outgrow supply, with doom-and-gloom consequences. Do some research about Malthus to answer these questions.
- What societal changes did Malthus propose to avoid the doom-and-gloom outcome he was predicting?
- Why do you think his predictions did not occur?
- What are the similarities and differences between Malthus’s theory and the logistic growth model?

## 5.5 Chapter Exercises

*You’ve got to be kidding me*, Fatima thought to herself as she received the e-mail from her boss. She reread the e-mail hoping that something would change on the screen: “Fatima, I need you to prepare a presentation on what our company has done in the past year for Mrs. Jorgensen. She’s old, keep it simple. Leave out any of the complex material because it will probably just bore her anyways.—John.”

Fatima joined R & R Consulting right after Anthony Jorgensen, the founder and CEO, had passed away. While Penelope Jorgensen inherited the major stake in the firm and was still listed as the firm’s CEO, the day-to-day running of operations was given to John Preston, the chief operating officer.

Fatima stared at her screen and wondered to what extent she should follow John’s advice and “keep it simple.” She’d only met Mrs. Jorgensen twice, but she’d always seemed to be pretty knowledgeable about the inner workings of the firm. Sure Mrs. Jorgensen wasn’t an expert in the field, but should she be treated like a helpless little old lady? *Not only is that sexist, it’s completely ageist! On the other hand, John’s words may have been chosen poorly, but maybe all Mrs. Jorgensen really wanted was a quick snapshot of what’s going on here*?

Fatima sat in silence for a few minutes, opened up PowerPoint, and just stared at her monitor trying to figure out the best way to proceed.

## Build Serious Strength and Size with a 5x5 Workout Program

The 5x5 program can't be beat for increasing strength, breaking plateaus, and adding muscle. Try out this classic lifting routine and see results!

One of the most popular muscle and strength-building programs of all time is the 5x5 protocol. Lifters were swearing by it 50 years ago, and in recent years, it has only gotten more popular. Why? Because it's simple, sustainable, and it straight-up works.

But there's a catch: 5x5 works best as a long-term commitment of at least a month or two, not a once-in-a-while thing. That's why it's best to follow a classic program, like Bodybuilding.com's 5x5 for Muscle and Strength.

Want to learn more? Here's what you need to know about 5x5 training.

### What Is 5x5 Weight Training?

Rather than doing a lot of different movements for each muscle group, 5x5 training hits just a few movements hard with 5 sets of 5 reps, sometimes followed by optional higher-rep accessory work. Here are two common protocols to alternate:

#### Workout 1

#### Workout 2

Some protocols would just have you alternate those two workouts in an A/B/A, B/A/B style. Others, like 5x5 for Muscle and Strength in BodyFit, add a third variation build around a third set of lifts, like box squats and incline presses.

### Is 5x5 Good for Building Muscle?

This training approach is designed to hit your muscles hard three times per week, then allow enough time to promote growth and recovery. It has been most popular among strength-seeking lifters, because it has you training in the strength-building sweet spot of 70-80 percent of your one-rep max (1RM) on big lifts. It's a great idea to use a 1RM calculator to find out your percentages.

But this is more than just a strength program. A well-designed 5x5 will reliably produce an increase in muscle mass, as well, provided you are eating enough calories to support this muscle growth. This version of 5x5 also contains some higher-rep accessory work that will help add muscle mass along with strength.

### Is 5x5 for Beginners?

Sure, it can be. But while 5x5 is sometimes advised as a beginner program, it's probably best used by intermediate lifters who have a base of training. No, you won't be maxing out, but the intensity and volume are higher than many bodybuilding-focused programs.

For many beginners, it's better to start off with a program built around sets of 8-10 reps, like Strength and Muscle for Beginners.

### How Heavy Should I Lift in 5x5 Workouts?

The biggest mistake people make in 5x5 training is going too heavy.

During the first week of the training program, you should err on the side of caution and use weights that feel light enough to allow you to hit every rep without much of a struggle. If you know what your one-rep max on the big lifts is, start 5x5 with around 65 percent of that max. After the first week, bump the weight up 5-10 pounds for the following week, and again the week after.

That may not sound like much, but it's enough to keep you progressing! Remember, 5x5 works when you hit all your reps, not when you miss them. Failure has no place in this plan!

### How Long Should You Do a 5x5 Program?

Stick with 5x5 training for at least four weeks. If you hit all your reps, you should hopefully be setting new rep maxes and seeing a significant increase in both strength and muscle.

If you're still making gains, stick with it for another four weeks. After that, it's a great time to switch the movements, or try another workout program.

### What's the Best Nutrition for 5x5 Training?

The classic 5x5 workouts work best when you are eating adequate calories. Using a calorie calculator to make sure you're in a slight caloric surplus is a great idea while following a 5x5 program.

As with any systematic, heavy lifting program, you need to ensure you have proper pre- and post-workout nutrition while doing 5x5 training. You may not get very sweaty doing sets of 5, but rest assured, your body is working hard!

The workouts are going to take a lot out of your body, so providing them with the nutrients to fuel the activity as well as recovery from it is essential. Aim for a gram of protein per pound of body weight each day, and don't be afraid to see the number on the scale go up along with the weight on the bar.

**5x5 and protein are a match made in gains heaven. Hit your reps and earn your shake!**

### A Timeless Approach for Strength and Size

If you can set your ego aside and do the work, you will join the legion of 5x5 believers on BodySpace and in the Bodybuilding.com forums. Then, you can always return to it in the future whenever you want to see the weight on the bar go up!

### About the Author

#### Shannon Clark

Shannon Clark is a freelance health and fitness writer located in Edmonton, Alberta, Canada.

## What Exercises Are Used in 5×5 Workouts?

Five by five is typically done on classic multi-joint barbell lifts, such as squats, deadlifts, presses, and rows—exercises that activate great amounts of muscle and work big areas of the body. **You shouldn’t attempt 5×5 with isolation movements such as curls, lateral raises, or leg extensions,** as those exercises don’t lend themselves to the use of heavy weights. Trying to curl a load that feels heavy for five reps can easily result in sore elbows or torn biceps over time, so restrict the 5×5 method to your main lift of the workout. Also, as 5×5 is strenuous and can be time consuming, don’t use it for more than three exercises in any one workout.

## Advanced 5 x 5 Workout Plans (3 Plans)

The problem with the basic 5 x 5 workout is there’s no focus on **hypertrophy (aka** *building muscle size and getting that pump)*. That’s ok because I’m about to solve that for you.

I’m going to give you **3 workout plans** you can follow. They’re all what I consider **advanced 5 x 5 workouts**.

In short, the 5 x 5 method will be used on your core compound exercises. But you’re also going to be integrating more exercises that focus on specific muscle development.

There are 2 simple goals with these workouts:

My suggestion is to follow the below workout schedule, which will make this a full 12-week **workout program**.

**Weeks 1 -4**: Advanced 5×5 Workout 1**Weeks 5-8**: Advanced 5×5 Workout 2**Weeks 9-12**: Advanced 5×5 Workout 3

Now for the 3 workout plans…

### Advanced 5×5 Workout 1: Strength and Hypertrophy

This first **advanced 5 x 5 workout plan is based on strength and hypertrophy**. The workout schedule is similar to the basic 5×5 workout. Think *‘strong and jacked!’*

The two major differences are:

- You’ll be doing a few more exercises catered to building muscle size
- For the hypertrophy part of the workouts, you will be supersetting two exercises
*(going from one exercise to another with no rest between that set = a superset)* - You’re going to be in the gym longer, so I suggest drinking BCAAs during these workouts
*(read about***my favorite BCAA supplement here**)

Similar to the basic 5 x 5 workout, you’ll have 2 workouts that you’ll alternate *(noted as A and B below)*. So a Monday – Wednesday – Friday schedule will look like…

**Monday:**Workout A**Wednesday:**Workout B**Friday:**Workout A

*The following Monday you would start with Workout B and continue alternating accordingly.*

#### Strength and Hypertrophy Workouts

#### Workout A

Exercises | Sets x Reps |
---|---|

Squats | 5 x 5 |

Barbell Rows | 5 x 5 |

Bench Press | 5 x 5 |

Leg Extension – superset with below | 3 x 8 |

Leg Curls | 3 x 8 |

Dumbbell Curls – superset with below | 3 x 10 |

Rope Pressdowns | 3 x 10 |

#### Workout B

Exercises | Sets x Reps |
---|---|

Squats | 5 x 5 |

Overhead Press | 5 x 5 |

Deadlifts | 5 x 5 |

Seated Rows – superset with below | 3 x 8 |

Dumbbell Flyes | 3 x 8 |

Pull-ups – superset with below | 3 x 10 |

Dips | 3 x 10 |

### Advanced 5×5 Workout 2: Bodybuilding Focused

In this workout plan, you’re going to truly merge the 5 x 5 strength part with **bodybuilding workout**. If you’re like me, you want that extra volume and walk away with that pump *(like so pumped you can’t fit through the door…lol!)*.

In fact, this is the perfect type of workout for building mass. And here’s why:

- You’re building strength at the beginning of the workout with your compound exercises
- You’re following that up with more training volume
*(sets and reps)*to build muscle - You’ll be doing some isolation exercise to bring out those striations and definition
- Each workout focuses on a specific muscle group

This is a ‘**get huge, get strong, and get jacked**‘ workout!

In this workout plan, you’ll be training 4 days a week. Here’s a schedule you can use:

**Monday****:**Workout 1 – Back**Tuesday****:**Workout 2 – Check and Biceps**Thursday:**Workout 3 – Legs**Friday**: Workout 4 – Shoulders and Triceps

You can arrange to cater to your schedule, but make sure you throw in a rest day after two workout days.

Workouts 1 and 3 *(in the case above, Monday and Thursday)* will be a little more taxing on your central nervous system as you’re doing 5×5 for 2 compound lifts. So make sure you eat well the day before if you train early mornings, and that day if you’re training in the evenings.

Lastly, you’re only doing the 5×5 method for each compound lift once a week except for deadlifts. I have you doing deadlifts on back and leg day *(this is a common workout for me!)*. You can change this around if you want to focus on another compound lift.

#### Bodybuilding Focused Workouts

#### Workout 1: Back

Exercises | Sets x Reps |
---|---|

Barbell Rows | 5 x 5 |

Deadlifts | 5 x 5 |

Dumbbell Rows | 3 x 8 |

Seated Rows | 3 x 8 |

Lat Pulldowns | 3 x 10 |

#### Workout 2: Chest and Biceps

Exercises | Sets x Reps |
---|---|

Bench Press | 5 x 5 |

Incline Bench Press | 4 x 6 |

Dumbbell Press | 3 x 8 |

Incline Dumbbell Flyes | 3 x 8 |

Barbell (or EZ Bar) Curls | 3 x 10 |

Dumbbell Hammer Curls | 3 x 10 |

#### Workout 3: Legs

Exercises | Sets x Reps |
---|---|

Squats | 5 x 5 |

Deadlifts | 5 x 5 |

Leg Press | 4 x 12 |

Leg Extensions | 3 x 12 |

Leg Curls | 3 x 12 |

Stiff-leg Deadlifts | 3 x 12 |

#### Workout 4: Shoulders and Triceps

Exercises | Sets x Reps |
---|---|

Overhead Press | 5 x 5 |

Seated Dumbbell Press | 4 x 6 |

Lateral Raises | 3 x 8 |

Bent-over Raises | 3 x 8 |

Skull-crushers | 3 x 10 |

Rope Pressdowns | 3 x 10 |

***If you’re interested in more bodybuilding workouts, then check out my 8 Week Bodybuilding Workout Plan post.*

### Advanced 5×5 Workout 3: Strength and Conditioning

This will probably be the most **unique and challenging workout plan** you’ve ever tried. I’ll admit I was a bit intimidated by it when I first tried *(like ‘pull the covers over my head so the ghosts don’t exist’ intimidated!)*.

But since, it’s become one of my favorite types of workouts, and you’re going to see why once you try it!

Of course, 5×5 will be part of the program but in an unnatural way. You’re going to think I’m crazy here but you’re going to be s**upersetting heavy compound exercises**.

I know I know…you’re thinking *WTF??*

Stay with me. Because the strength and muscle gains are going to be accompanied by getting your body in extreme condition! And you’re going to amaze yourself by learning what you’re truly capable of.

You’ll be training 4 days a week and your workout schedule will look like this:

**Monday**: Lower Body and Core**Tuesday**: Upper Body**Thursday**: Lower Body and Core**Friday**: Upper Body

#### What’s So Special About Workout 3?

Have you ever heard the terms **agonist-antagonist muscles**? If so, then skip ahead to the workouts below. If not, let me explain…

The best way to describe an agonist-antagonist workout is by combining push-pull. An example is doing a set of barbell rows then supersetting that with a set of bench press.

Let me further explain. 1 set of barbell rows followed by 1 set of bench press equals 1 superset. At that point, you can rest before doing another superset. You’ll do 5 sets of 5 reps *(5 supersets, and 5 reps of each exercise during each superset)*.

Your lower body days will be a little different as you’ll be super setting with an abdominal exercise with your compound lower body/leg exercises *(the reps for abs will be increased)*.

I assure you that **you can do this**! Is it easy? Hell no! But it’s totally worth the gains you’ll get from it!

***If you want to learn more about agonist-antagonist training, I have an entire post (with full workout program) dedicated to it called Lean and Strong Workout Program Using Agonist-Antagonist Supersets.*

#### Strength and Conditioning Workouts

#### Workout 1: Lower Body

Exercises | Sets x Reps |
---|---|

Squats – superset with below | 5 x 5 |

Leg/Knee Raises | 5 x 10 |

Leg Press – superset with below | 4 x 12 |

Crunches or Decline Sit-ups | 4 x 12 |

Leg Extensions – superset with below | 4 x 12 |

Leg Curls | 4 x 12 |

#### Workout 2: Upper Body

Exercises | Sets x Reps |
---|---|

Barbell Rows – superset with below | 5 x 5 |

Bench Press | 5 x 5 |

Seated Rows – superset with below | 4 x 8 |

Incline Bench Press | 4 x 8 |

Barbell Curls – superset with below | 4 x 10 |

Skull-crushers | 4 x 10 |

#### Workout 3: Lower Body

Exercises | Sets x Reps |
---|---|

Deadlifts – superset with below | 5 x 5 |

Leg/Knee Raises | 5 x 10 |

Front Squats – superset with below | 4 x 12 |

Crunches or Decline Sit-ups | 4 x 12 |

Leg Extensions | 4 x 12 |

Good Mornings – superset with below | 4 x 12 |

#### Workout 4: Upper Body

Exercises | Sets x Reps |
---|---|

Overhead Press – superset with below | 5 x 5 |

Barbell Rows | 5 x 5 |

Seated Dumbbell Press – superset with below | 4 x 8 |

Lat Pulldowns | 4 x 8 |

Close-grip Bench Press – superset with below | 4 x 10 |

Preacher Curls | 4 x 10 |

## 5x5 Full-Blown

A classic muscle- and strength-building training protocol gets a considerable upgrade with this 5-day full-body routine.

Even the best, most tried-and-true training techniques can use a tweak every now and then. The classic 5x5 scheme is no exception. It&rsquos always produced great results in the gym, but now I&rsquom helping it evolve for even greater gains.

The 5x5 protocol has been around for at least a half century, so it&rsquos nothing new. Iconic bodybuilder Reg Park, a three-time Mr. Universe in the 1950s and 60s and Arnold Schwarzenegger&rsquos boyhood idol, swore by the method and is considered a pioneer of 5x5 training.

The method is what it sounds like: 5 sets of 5 reps on a given exercise. The 5-rep count is something of a &ldquotweener&rdquo that falls within the well-established 8-12-rep range ideal for building size and the 1-3-rep range favored by powerlifters for maximizing pure strength.

So the question is: Are sets of five reps better for size or strength? Through anecdotal and empirical research, the answer is unequivocally both. And with my version of 5x5, you&rsquoll be promoting muscle-building even more (I&rsquoll explain in a minute).

A typical 5x5 program entails using a relatively heavy weight and taking ample rest periods between sets (2-3 minutes or more). And these workouts are nearly always done as bodypart split routines &ndash ranging from upper body-lower body splits to 3-5-day splits.

My twist on 5x5 keeps the weight heavy like the others, but differs in two key areas:

(1) It consists of full-body workouts, hitting each major muscle group every day and (2) it cuts down on the rest, where you&rsquoll essentially compete against yourself to complete your 25 total reps (5 sets, 5 reps) per exercise in less time each subsequent workout.

The first added element will maximize fat loss, as research shows that full-body training is better for burning body fat than split-body routines. The second aspect, as I mentioned above, will systematically increase intensity to help you pack on more muscle mass.

Sound challenging? Good, that&rsquos the point!

### Full-Blown Breakdown

My 5x5 routine consists of (you guessed it) five workouts. As with my other recent full-body programs, you&rsquoll be doing 10 exercises in each session &ndash one move per bodypart (chest, back, legs, shoulders, traps, biceps, triceps, forearms, calves, abs).

The one major difference between this program and my others is that you&rsquoll do the SAME EXERCISES in all five workouts. And you&rsquoll use the same weight on each of those 10 moves as well.

Same exercises, same weights, same number of sets and reps per exercise (5x5). Exact same workout for five straight days? Nope. One variable will change daily&hellip

Every workout, your goal will be to complete the 5 sets of 5 reps for each exercise in less time. If and when you&rsquore able to accomplish this, it means your training intensity will be increasing from workout to workout. It&rsquos a very simple formula: The same amount of work performed (same sets and reps, same amount of weight) in less time equals a more intense workout.

### Workout Rundown

In your first of five workouts (Workout 1), you&rsquoll select your 10 exercises &ndash one for each muscle group. The exercises I chose are shown below. You can either do those same ones, pick your own exercises, or a mix of both (do some of the same moves I did and then pick others yourself). It&rsquos up to you.

You can choose pretty much any type of exercises you want &ndash free weight moves, machines, cables, kettlebells, Smith machine, etc. I recommend selecting compound (multi-joint) movements for large bodyparts &ndash chest, back, legs and shoulders &ndash but you certainly don&rsquot have to. If you&rsquod rather do cable crossovers than dumbbell bench press for five days, knock yourself out.

That said, you&rsquoll need to pick exercises you can go sufficiently heavy on, since you&rsquoll only be doing five reps per set. In other words, bodyweight crunches and sit-ups are probably not your best options for ab exercises go with something weighted instead, like Smith machine crunches or cable crunches.

Once you know the 10 exercises you&rsquore going to do, head to the gym for Workout 1. But don&rsquot forget to bring a timer/stopwatch and something to log your weights and times with &ndash a training log, a spiral notebook, your cell phone, etc.

Here are the exercises I picked:

- Bench Press
- Barbell Row
- Squat
- Shoulder Press
- Shrug
- Standing Calf
- Close-Grip Bench Press
- Barbell Curl
- Wrist Curl
- Crunch

Weight selection is sort of hit or miss on this program. Clearly, you don't want to use your true 5-rep max (5RM). There's no way to complete 5 reps with your max weight for 5 reps on five successive sets, no matter how long you rest. So you need to undershoot the weight a bit. Given that workout 1 starts with only one minute of rest between sets, and each workout the rest gets shorter, you'll definitely need to go much lighter than your 5-rep max.

How light depends on how fast your body can recover in one minute between 5 successive sets. Research shows that females recover quicker between sets and complete more reps on successive sets of an exercise than men. Most people will find that somewhere between their 8-10-rep max is perfect. If it's too easy in Workout 1, you can increase the weight while decreasing rest time in Workout 2. And same thing if you overshoot the weight in workout 1 and can't complete all 5 reps on all 5 sets simply reduce the weight as you reduce the time in Workout 2.

Record that weight in your training log or phone so you remember how much weight to use on that exercise the next four days.

Once you've got your weight picked out and are ready to go, start your timer, then do 5 sets of 5 reps with that weight using 1-minute rest periods. After your last set, record how long it took you to do all 25 reps. It should take you somewhere around 5-6 minutes to compete all 5 sets of 5 reps, so Workout 1 should take just under an hour. Each subsequent workout will get shorter.

Once you&rsquove recorded your time, go right to the next exercise and repeat the process. Do this for all 10 exercises.

The next day (Workout 2), you&rsquoll be doing what looks on paper like the same workout &ndash but here, your goal will be to beat your previous completion times on all exercises. So if it took you, say, 5:56 (five minutes, 56 seconds) to do 5 sets of 5 reps on rows, shorten your rest periods by 5-10 seconds to knock at least 20-40 seconds off that time. Whatever the time was, jot it down.

Do the same thing the next three days (Workouts 3-5) &ndash the same 10 exercises, the same weights, and the same 5x5 set/rep scheme, but trying to beat the previous day&rsquos completion time on each move.

I designed this program to be done on five consecutive days (ie, Monday-Friday), but feel free to stretch it out longer than that by inserting rest days between workouts if you&rsquore a beginner or intermediate and think you need the additional recovery time. Make the program work for your schedule. If you&rsquore only able to get to the gym three days a week, do Workouts 1-3 this week and Workouts 4 and 5 next week, then start another routine to finish your week.

I posted to the below workouts, times and comments on my social media platforms while doing the 5x5 Full-Blown program, but you can do them whenever you want now.

### Workout 1

Feel free to choose your own exercises for each of the 10 muscle groups listed below.

Don't forget to record the weights you're using you'll use those same weights on the same exercises for all five workouts. And don't forget to log the time it took you to complete your 25 reps per exercise &ndash that's the variable that will change from day to day.

As you'll see, I'm including my time for each exercise, but that's just for an example. Don't feel like you have to hit my times. We're all at different points along our training spectrums, so don't worry about my times too much.

- Bench Press (5:45)
- Barbell Row (5:10)
- Squat (6:20)
- Shoulder Press (5:15)
- Shrug (5:20)
- Standing Calf (5:15)
- Close-Grip Bench Press (5:30)
- Barbell Curl (5:45)
- Wrist Curl (5:10)
- Crunch (5:10)

### Workout 2

My second time through the 5x5 Full-Blown workout, I beat all my times from Workout 1 by at least 10 seconds.

- Bench Press (5:15)
- Barbell Row (5:00)
- Squat (5:45)
- Shoulder Press (4:50)
- Shrug (5:00)
- Standing Calf (5:00)
- Close-Grip Bench Press (5:00)
- Barbell Curl (5:25)
- Wrist Curl (5:00)
- Crunch (4:50)

## 5.5: Exercises

**This is the first of six charts describing your physical capacity rating scale. Everyone is advised to begin the 5BX program with this chart at the lowest level, D-. Review the section "How to Begin" before starting with this chart.**

Age Groups

6 yrs level B : 7 yrs level A

**5BX Chart One Level A**

**Minutes in each exercise**

1. Feet astride, arms upward. Forward bend to floor touching then stretch upward and backward bend. Do not strain to keep knees straight.

2. Back lying, feet 6" apart, arms at sides. Sit up just far enough to see your heels. Keep legs straight, head and shoulders must clear the floor.

3. Front lying, palms placed under the thighs. Raise head and one leg, repeat using legs alternately. Keep leg straight at the knee, thighs must clear the palms. (Count one each time second leg touches floor.)

4. Front lying, hands under the shoulders, palms flat on the floor. Straighten arms lifting upper body, keeping the knees on the floor. Bend arms to lower body. Keep body straight from the knees, arms must be fully extended, chest must touch floor to complete one movement.

5. Stationary run - (count a step each time left foot touches floor. Lift feet approximately 4 inches off floor). Every 75 steps do 10 "scissor jumps". Repeat this sequence until required number of steps is completed.

Scissor jumps - Stand with right leg and left arm extended forward, and left leg and right arm extended backward. Jump up - change position of arms and legs before landing. Repeat (arms shoulder high).

## 5 Workout Program – Arnold’s Favourite Training Routine

The 5 workout program was popularised by the Austrian-American bodybuilding legend Arnold Schwarzenegger.

It serves as a proven training program for building muscle mass and strength in bodybuilders, as well as powerlifters to this day!

The 5 workout originated in the 1940’s. Arnold Schwarzenegger talks about it in his best selling book ”The Encyclopedia of Modern Bodybuilding.”

He says that he first learned about the 5 workout program from his bodybuilding idol Reg Park, who won Mr. Britain 1949 and Mr. Universe 1951.

Over time it has become one of the most used workouts for building strength and muscle mass.

Read: Build Your Shoulders With This Reg Park Workout For MassBoth, professional and amateur bodybuilders and powerlifters all swear by it – and rightfully so! The 5 workout is specifically designed to hit all of the major muscle groups hard, multiple times per week.

In his book The Education of a Bodybuilder (1977), Arnold Schwarzenegger said this about the 5 workout program:

From the beginning I was a big believer in the basic movements, because that was Reg Park’s preference. He would stay with the basic exercises – bench presses, chin-ups, squats, rowing, barbell curls, wrist curls, pullovers, leg extensions, calf raises. These were the movements that worked most directly on all body parts. I was following his example to the letter. And as it turned out, I could hardly have chosen more wisely. The basic exercises were creating for me a rugged foundation, a core of muscle I could later build upon for a winning body. Reg Park’s theory was that first you have to build the mass and then chisel it down to get the quality.

### 5 Workout

As the name suggests, the 5 workout program involves doing **5 sets of 5 repetitions on basic compound exercises**, such as bench press, barbell rows and squats, among others. This workout is usually done **3 times per week**, but there are many different workout variations where training can be done more or less often. The 5 program can be done as a full body workout or as a split workout routine.

Can the 5 workout be done by beginners? Absolutely. However first, you must make sure that you have learned proper form for all of the big compound exercises used in this workout program. This will reduce the risk of sustaining an injury.

The 5 workout was created around the principles of progressive overload. This means that the exercise load changes as you progress. You have to progressively increase your weights, which will trigger new muscle and strength gains.

Progressive overload is perfectly described by the legend of an Ancient Greek wrestler and Olympic victor called Milo of Croton. He’s described as having near-superhuman strength. According to the legend, while training for the Olympic games he would lift a calf on his shoulder and carry it every day. As the calf grew over time, so did his strength and muscle.

The lesson here is that he started light and added more weight gradually, which kept making him stronger throughout his training.

### 5 Workout Plan (Full Body)

Lifting heavy weights is the cornerstone of the 5 workout program, but make sure that you don’t go too heavy too early.

During your first 2 weeks it is recommended that you start with weights around **65% of your 1RM**. This will allow your muscles to get accustomed to the weight and reduce the risk of injury.

#### Day 1:

** Bench Press** (5 reps x 5 sets @ 65% 1RM)

**(5 reps x 5 sets @ 65% 1RM)**

*Barbell Rows***(5 reps x 5 sets @ 65% 1RM)**

*Squat*#### Day 3:

** Weighted Dips** (5 reps x 5 sets @ 65% 1RM)

**(5 reps x 5 sets @ 65% 1RM)**

*Weighted Chin-Ups***(5 reps x 5 sets @ 65% 1RM)**

*Deadlifts*#### Day 5:

** Incline Bench Press** (5 reps x 5 sets @ 65% 1RM)

**(5 reps x 5 sets @ 65% 1RM)**

*Front Squat***(5 reps x 5 sets @ 65% 1RM)**

*Barbell Rows*You may increase the weight every 2 weeks by 5-10%.

### 5 Workout Plan (Upper/Lower Split)

Alternatively, if you prefer to train using an upper/lower body split, try this 5 workout program instead:

#### Day 1:

** Bench Press** (5 reps x 5 sets @ 65% 1RM)

**(5 reps x 5 sets @ 65% 1RM)**

*Barbell rows*#### Day 3:

** Squat** (5 reps x 5 sets @ 65% 1RM)

**(5 reps x 5 sets @ 65% 1RM)**

*Stiff Leg Deadlift***(5 reps x 5 sets @ 65% 1RM)**

*Calf Raise*#### Day 5:

** Incline Bench Press** (5 reps x 5 sets @ 65% 1RM)

**(5 reps x 5 sets @ 65% 1RM)**

*Wide Grip Pull-Up*Remember about progressive overload? Make sure to increase the weights every 2 weeks. You can add 5-10% of weight to each exercise.

By the time you get to week 12 you should be comfortably hitting new 1RM’s. Keep in mind that nutrition is equally as important as training, if not more. So if you want to keep gaining new muscle you must up your calorie intake – especially your protein.

If you find that you are struggling with increasing your weight every 2 weeks, just stick with the same weight for an extra week!

#### Key points:

- You have implement progressive overload. Try to increase the weights by 5% every week or 10% every 2 weeks.
- Don’t start with heavy weights in the first 2 weeks. You will be increasing the weight every 2 weeks, so you want to leave some room.
- Eat and rest well!
- You should de-load or take a week off when you start feeling beat up. This can happen after 5 to 8 weeks of training.

### 5 Workout F.A.Q

The 5 workout is a training program popularised by the legendary bodybuilder Arnold Schwarzenegger. It is a proven training method for building muscle size and strength in powerlifters, as well as bodybuilders.

The 5 workout designed to hit all of the major muscle groups hard, multiple times per week. It is an effective workout for building muscle size and strength.

The 5 workout has been used by legendary bodybuilders like Arnold Schwarzenegger and Reg Park to build muscle mass and strength. This workout is built around basic compound exercises, such as bench press, barbell rows and squats, which target many muscle groups at once.

By doing the 5 workout with progressive overload you should be able to add 14-28kg (30-60lbs) to your compound lifts (bench press, squats, deadlift) after 12 weeks of training.

## 5 X 5 Weight Training Program

The 5 x 5 weight training program is a simple and complete power training program.

For those of you who are in the intermediate to advanced levels of training, this program may help you add more power to your body.

The 5 x 5 training style has been around for years and was quite popular in the early 60’s through 70’s, being used by top body builders and strength athletes.

Today, it is an often forgotten form of training but it still remains one of the most effective ways to add power and muscle mass to the body.

If your looking to add more power and size to your body, this type of training is for you. However, this is a tough program that places a tremendous amount of stress to your body. It is for this reason that you cannot do a 5 x 5 program for any longer than 10 to 12 weeks at a time. Once the 5 x 5 training cycle has been completed, you need to take at least 2 weeks off from training to allow your joints, tendons, and muscles to repair.

Alright, what is a 5 x 5 training program all about? The program is all about doing 5 sets of 5 repetitions with a heavy, working weight. For example, let’s say I want to do a 5 x 5 for the bench press. I would first do two sets of 20 repetitions for my warm up and than I would load up the bar with my working weight that would allow me to do 5 sets of 5 repetitions.

Let’s say my progression is as follows for a 250 pound max bench press:

warm up sets: 2 x 20 repetitions with 135 pounds

working sets: 5 x 5 repetitions with 225 pounds

Once I can complete 5 repetitions for each set, I will increase the weight by about 10% for my next session. For this type of program, it is all about strength so you need to be fully recovered before you attempt the next set. It is for this reason that you must rest about 1 ½ minutes between sets.

The thing you have to remember about this program is that it is all about compound exercises. You won’t be doing 5 x 5 for cable cross overs – You just won’t get the same benefit. You need to pack on the weight and do 5 x 5 for compound movements such as the squat and dead lift. If you can do this type of training for these kinds of exercises, be prepared to pack on some serious beef!

You can add a couple of isolation exercises but no more than one to two per session.

Personally, I prefer to do 5 x 5 training for my whole body. That is, I prefer to do full body workouts for my 5 x 5 sessions. For some reason, I get the most out of doing full body, high intensity workouts. Also, you need to break up your exercises because you will open yourself up to overuse injuries that can hinder your progress.

A heavy bench press is enough for one week so to break up the routine, we can use the incline press on the next training session (see below). This is very important because doing two heavy sessions for the same exercise places too much stress on the body.

Now, to get the benefit out of this kind of program, you absolutely must, must have a complete nutritional program down. If you don’t, your going to open yourself up to injuries. Make sure you have a lot of nutrients in your body before you attempt to do this kind of training. Go to the diet page to find out more.

## Model solution 4

This is one of those problems where it is much easier to use the well-ordering principle (which is equivalent to induction but doesn’t appear until later in this book). The proof of equivalence between mathematical induction and the well-ordering principle does not require any results from analysis, so there is no circularity if we use the well-ordering principle. I’ll show it here so you get an idea of how clean the solution can be (compared to the above three solutions). I will try to remember to point back to here when I get to the well-ordering principle in the book.

This set is non-empty since . Also, is a lower bound for since for all . Thus by the well-ordering principle there is a minimum element . By definition of minimum element, so is an upper bound for . If were an upper bound for then since is not there is some such that . Thus . We also have so . Thus , which shows that . But this contradicts the fact that is the minimum element of .