2.6: More on Series - Mathematics

2.6: More on Series - Mathematics

Geometric Sequence Calculator

Calculate anything and everything about a geometric progression with our geometric sequence calculator. This geometric series calculator will help you understand the geometric sequence definition so you could answer the question what is a geometric sequence? We explain the difference between both geometric sequence equations, the explicit and recursive formula for a geometric sequence, and how to use the geometric sequence formula with some interesting geometric sequence examples.

We also have built a "geometric series calculator" function that will evaluate the sum of a geometric sequence starting from the explicit formula for a geometric sequence and building, step by step, towards the geometric series formula.


In this section, you will learn GEMDAS rule that  can be used to simplify or evaluate complicated numerical expressions with more than one binary operation easily.

Very simply way to remember  GEMDAS rule :

1. In a particular simplification, if you have both  multiplication and division, do the operations one by one in the order from left to right.

2. Multiplication does not always come before division. We have to do one by one in the order from left to right. 

3. In a particular simplification, if you have both  addition and subtraction, do the operations one by one in the order from left to right.

In the above simplification, we have both division and multiplication. From left to right, we have division first and multiplication next.

So we do division first and multiplication next.

To have better understanding on PEMDAS rule, let us look at some practice problems. 

2.6: More on Series - Mathematics

Department of Mathematics
College of Science & Engineering

The 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC'10) will take place at San Francisco State University, August 2-6, 2010.

Registration and all the talks and posters will take place in Jack Adams Hall at San Francisco State's Cesar Chavez Student Center (coordinates H7 on the campus map). The conference will start on Monday, August 2, at 8:45 a.m. (registration opens at 8:00 a.m.). Please join us for an informal opening reception on Sunday, August 1, at 5:00 p.m. (follow the signs in the Student Center).

The conference will feature invited lectures, contributed presentations, poster session, and software demonstrations. As usual, there will be no parallel sessions.

Topics include all aspects of combinatorics and their relations with other parts of mathematics, physics, computer science, and biology.

The official languages of the conference are English, French, and Spanish.

Main Lesson: Order of Operations

Some people use PEMDAS or "Please Excuse My Dear Aunt Sally" to remember the order of operations.
P=parentheses (and other grouping symbols)
E=exponents (introduced in 6th grade)
A =add

When mathematicians from around the world met long ago to decide on a standard order for doing mathematical operations, this is the order they agreed upon:

  1. Do any math inside grouping symbols first: Parentheses, brackets & braces.
  2. Evaluate numbers with exponents: Whole number exponents will be explained as part of 6th grade lessons. They are not included in this lesson, except to know the correct order.
  3. Multiplication or Division: Multiplying and dividing have the same priority. When you are reading from left to right, do whichever one you come to first. Skip adding and subtracting until after all multiplication and division has been done.
  4. Addition or Subtraction: Adding and subtracting have the same priority. When you are reading from left to right, do whichever one you come to first.

Have your children work through these problems to practice applying the order of operation rules. After these model problems, there some more practice problems for your children to try on their own.

Practice Using Order of Operations

Parentheses come first, so 7 + 6 = 13 .

Plug in the 13 where (7 + 6) was, so: 14 - 13

There are no exponents in this lesson, so go on to operations.

This problem contains only subtraction, so subtract.
14 - 13 = 1

Parentheses come first, so 8 - 4 = 4 .

Plug in the 4 where (8 - 4) was, so: 4 + 5 x 8 .

There are no exponents in this lesson, so go on to operations.

This problem contains addition and multiplication. Multiplication comes before addition, so 5 x 8 = 40 .

That leaves 4 + 40. Finally, add 4 + 40 = 44 .

Children are often overwhelmed by a complicated math expression or equation. Remind them to focus on just one step at a time. Big tasks become easier to think about and accomplish when broken down into small steps.

Try evaluating these expressions by following the order of operations rules and then check your answers by clicking on the "Show/ Hide Answer" link.

Parentheses come first, so 3 - 1 = 2 .

Multiplication comes before addition, so 2 x 6 = 12

Addition is all that is left, so 4 + 12 = 16 .

Parentheses come first, so 11 + 9 = 20

Multiplication comes before subtraction, so 5 x 2 =10

Subtraction is all that is left, so 20 - 10 = 10

Grouping symbols come first, and parentheses before brackets, so 2 + 8 = 10 .

That leaves: [17-10+2] in brackets. 17-10 = 7 ,

The total value inside the brackets is 9.

That leaves just division: 9 รท 3 = 3


  • Mathematicians agreed upon certain rules for solving math, called the order of operations.
  • Grouping symbols come first. If there is more than one grouping symbol they go in this order: parentheses, brackets, braces.
  • Exponents come next. You will use these in future lessons. For this lesson, you just need to know that they come after grouping symbols in the order.
  • Multiplication or Division comes next. They have the same priority, so whichever is first left to right goes first.
  • Addition or Subtraction comes next. They have the same priority, so whichever is first left to right goes first.
  • Don't feel overwhelmed by complicated problems. Take them step by step, one piece at a time, and you'll be able to solve them!

Test Questions

Review the recap points above with your children and then print out the Assessment Worksheet below.

At least 7 out of 10 correct will show that your children are ready to go on to the next lesson: Writing Simple Expressions.

Types of Events That Influence Probability

Picking a card, tossing a coin, and rolling a dice are all random events. But in the study of probability, there are at least 3 types of events which impact outcome:


In this type of event, each occurrence is not influenced at all by other events.

An example is tossing a coin to get heads or tails. Each coin toss is an independent event, which means the previous coin tosses do not matter. The chances of getting heads or tails is 1/2 or 50% every time a coin is tossed. Likewise, each time dice is rolled whatever was rolled on the previous roll has no impact on subsequent rolls.


This is when the outcome is influenced by other events, also called &lsquoconditional&rsquo event. And if two events are dependent events, one event affects the probability of another event.

An example is drawing cards. Every time you take a card, the number of cards decrease (there are 52 cards in a deck), which means the probabilities change. For instance, the chance of getting a king is 4 out of 52 on your first draw. If you get a king on your first card, the second card will have a lower chance of being a king, and the probability becomes 3 out of 51.

Mutually Exclusive

These are events that cannot happen at the same time. One event occurs or the other, but never both. Examples of this include a coin toss and turning left or right.

In a deck of cards, aces and kings are mutually exclusive, because both group of cards are entirely different from each other. On the other hand, heart cards and kings are not mutually exclusive because the group includes the king of hearts.

Even and Odd Numbers

Even numbers are numbers that can be divided evenly by 2. Even numbers can be shown as a set like this:

Odd numbers are numbers that cannot be divided evenly by 2. Odd numbers can be shown as a set like this:

Zero is considered an even number.

Is It Even or Odd?

To tell whether a number is even or odd, look at the number in the ones place. That single number will tell you whether the entire number is odd or even.

Consider the number 3,842,917. It ends in 7, an odd number. Therefore, 3,842,917 is an odd number. Likewise, 8,322 is an even number because it ends in 2.

Adding Even and Odd Numbers

Subtracting Even and Odd Numbers

Multiplying Even and Odd Numbers

Division, or The Fraction Problem

As you can see, there are rules that tell what happens when you add, subtract, or multiply even and odd numbers. In any of these operations, you will always get a particular kind of whole number.

But when you divide numbers, something tricky can happen?you might be left with a fraction. Fractions are not even numbers or odd numbers, because they are not whole numbers. They are only parts of numbers, and can be written in different ways.

For example, you can't say that the fraction 1/3 is odd because the denominator is an odd number. You could just as well write that same fraction as 2/6, in which the denominator is an even number.

The terms ?even number? and ?odd number? are only used for whole numbers and their opposites (additive inverses).

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The Department offers instruction in Mathematics and Applied Mathematics at all levels - Graduate and Undergraduate.


The Department offers instruction in Mathematics at all levels. See the Undergraduate Program and Graduate Program pages for comprehensive information.

The Department of Mathematics  has established research groups in several areas of contemporary mathematics, including algebra, geometry, topology, analysis and noncommutative geometry. These research groups and their activities are described in more detail on the Department's Research page.

The Department is one of the three departments comprising the newly formed School of Mathematical and Statistical Sciences.   

The Department of Mathematics is one of two departments comprising the School of Mathematical and Statistical Sciences (SMSS).   

  • Fibonacci Numbers and Nature
    Fibonacci and the original problem about rabbits where the series first appears, the family trees of cows and bees, the golden ratio and the Fibonacci series, the Fibonacci Spiral and sea shell shapes, branching plants, flower petal and seeds, leaves and petal arrangements, on pineapples and in apples, pine cones and leaf arrangements. All involve the Fibonacci numbers - and here's how and why.
  • The Golden section in Nature
    Continuing the theme of the first page but with specific reference to why the golden section appears in nature. Now with a Geometer's Sketchpad dynamic demonstration.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

The Puzzling World of Fibonacci Numbers

  • The Easier Fibonacci Puzzles page
    has the Fibonacci numbers in brick wall patterns, Fibonacci bee lines, seating people in a row and the Fibonacci numbers again, giving change and a game with match sticks and even with electrical resistance and lots more puzzles all involve the Fibonacci numbers!
  • The Harder Fibonacci Puzzles page
    still has problems where the Fibonacci numbers are the answers - well, all but ONE, but WHICH one? If you know the Fibonacci Jigsaw puzzle where rearranging the 4 wedge-shaped pieces makes an additional square appear, did you know the same puzzle can be rearranged to make a different shape where a square now disappears?
    For these puzzles, I do not know of any simple explanations of why the Fibonacci numbers occur - and that's the real puzzle - can you supply a simple reason why??

The Intriguing Mathematical World of Fibonacci and Phi

  • The Mathematical Magic of the Fibonacci numbers
    looks at the patterns in the Fibonacci numbers themselves: the Fibonacci numbers in Pascal's Triangle using the Fibonacci series to generate all right-angled triangles with integers sides based on Pythagoras Theorem.
    • An auxiliary page:
      • More on Pythagorean triangles
      • The first 500 Fibonacci numbers.
        completely factorized up to Fib(300) and all the prime Fibonacci numbers are identified up to Fib(500).
      • A Formula for the Fibonacci numbers
        Is there a direct formula to compute Fib(n) just from n? Yes there is! This page shows several and why they involve Phi and phi - the golden section numbers.
      • Fibonacci bases and other ways of representing integers
        We use base 10 (decimal) for written numbers but computers use base 2 (binary). What happens if we use the Fibonacci numbers as the column headers?

      The Golden Section

      1� 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

      which we call Phi (note the capital P), the Greek letter &Phi , on these pages. The other number also called the golden section is Phi-1 or 0�. with exactly the same decimal fraction part as Phi. This value we call phi (with a small p), the Greek letter &phi here. Phi and phi have some interesting and unique properties such as 1/phi is the same as 1+phi=Phi.
      The third of Simon Singh's Five Numbers programmes broadcast on 13 March 2002 on BBC Radio 4 was all about the Golden Ratio. It is an excellent introduction to the golden section. I spoke on it about the occurrence in nature of the golden section and also the Change Puzzle.
      Hear the whole programme (14 minutes) using the free RealOne Player.

        The Golden section - the Number
        The golden section is also called the golden ratio, the golden mean and Phi.

      • Geometry and the Golden section or Fantastic Flat Facts about Phi
        See some of the unexpected places that the golden section (Phi) occurs in Geometry and in Trigonometry: pentagons and decagons, paper folding and Penrose Tilings where we phind phi phrequently!
        • An auxiliary page on Exact Trig Values for Simple Angles explores the many places that Phi and phi occur when we try to find the exact values of the sines, cosines and tangents of simple angles like 36° and 54°.
        • Phi's Fascinating Figures - the Golden Section number
          All the powers of Phi are just whole multiples of itself plus another whole number. Did you guess that these multiples and the whole numbers are, of course, the Fibonacci numbers again? Each power of Phi is the sum of the previous two - just like the Fibonacci numbers too.

            is an optional page that expands on the idea of a continued fraction (CF) introduced in the Phi's Fascinating Figures page.
        • There is also a Continued Fractions Converter (a web page - needs no downloads or special plug-is) to change decimal values, fractions and square-roots into and from CFs.
        • This page links to another auxiliary page on Simple Exact Trig values such as cos(60°)=1/2 and finds all simple angles with an exact trig expression, many of which involve Phi and phi.
        • The Golden String

          • Fibonacci Rabbit Sequence
            See show how the golden string arises directly from the Rabbit problem and also is used by computers when they compute the Fibonacci numbers. You can hear the Golden sequence as a sound track too.
            The Fibonacci Rabbit sequence is an example of a fractal - a mathematical object that contains the whole of itself within itself infinitely many times over.

          Fibonacci - the Man and His Times

          • Who was Fibonacci?
            Here is a brief biography of Fibonacci and his historical achievements in mathematics, and how he helped Europe replace the Roman numeral system with the "algorithms" that we use today.
            Also there is a guide to some memorials to Fibonacci to see in Pisa, Italy.

          More Applications of Fibonacci Numbers and Phi

          • The Fibonacci numbers in a formula for Pi ()
            There are several ways to compute pi (3� 26535 ..) accurately. One that has been used a lot is based on a nice formula for calculating which angle has a given tangent, discovered by James Gregory. His formula together with the Fibonacci numbers can be used to compute pi. This page introduces you to all these concepts from scratch.
          • Fibonacci Forgeries
            Sometimes we find series that for quite a few terms look exactly like the Fibonacci numbers, but, when we look a bit more closely, they aren't - they are Fibonacci Forgeries.
            Since we would not be telling the truth if we said they were the Fibonacci numbers, perhaps we should call them Fibonacci Fibs !!
          • The Lucas Numbers
            Here is a series that is very similar to the Fibonacci series, the Lucas series, but it starts with 2 and 1 instead of Fibonacci's 0 and 1. It sometimes pops up in the pages above so here we investigate it some more and discover its properties.
            It ends with a number trick which you can use "to impress your friends with your amazing calculating abilities" as the adverts say. It uses facts about the golden section and its relationship with the Fibonacci and Lucas numbers.
            • The first 200 Lucas numbers and their factors
              together with some suggestions for investigations you can do.

            2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 .

            Fibonacci and Phi in the Arts

            • Fibonacci Numbers and The Golden Section In Art, Architecture and Music
              The golden section has been used in many designs, from the ancient Parthenon in Athens (400BC) to Stradivari's violins. It was known to artists such as Leonardo da Vinci and musicians and composers, notably Bartók and Debussy. This is a different kind of page to those above, being concerned with speculations about where Fibonacci numbers and the golden section both do and do not occur in art, architecture and music. All the other pages are factual and verifiable - the material here is a often a matter of opinion. What do you think?


            • Fibonacci and Phi Formulae
              A reference page of about 350 formulae and equations showing the properties of the Fibonacci and Lucas series, the general Fibonacci G series and Phi also available as
            • Linear Recurrence Relations and Generating Functions
              An experimental page to list as many (linear) reurrence relations as I can find on ALL of the topics on these pages: Common Number series, Fibonacci Numbers and their many Generalisations, Pythagorean Triangles, Continued Fractions, Polygonal Numbers (ordinary and central), Egyptian Fractions.
            • Links and Bibliography
              Links to other sites on Fibonacci numbers and the Golden section together with references to books and articles.

            Awards for this WWW site

            This site went live in March 1996 and is therefore the oldest maths site on the web!
            Hosted by the Department of Mathematics, Surrey University, Guildford, UK, where the author was a Lecturer in the Mathematics and Computing departments 1979-1998.

            Watch the video: Creating an Advanced Face Rig Pt. 2 Blender Tutorial (November 2021).