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Geometric progression


We can define geometric progression, or simply P.G., as a succession of real numbers obtained, except for the first, by multiplying the previous number by a fixed quantity what, call reason.

We can calculate the ratio of progression if it is not sufficiently evident by dividing two consecutive terms.

For example, in succession (1, 2, 4, 8,…) we have q = 2.

General term calculation

In a geometric progression of reason what, terms are obtained, by definition, from the first, as follows:

The1The2The3The20Theno
The1The1xqThe1xq2The1xq19The1xqn-1

Thus we can deduce the following expression from the general term, also called the nth term, for any geometric progression.

Theno = a1 . whatn-1

So if for example The1 = 2 and what = 1/2, then:

Theno = 2 . (1/2)n-1

If we want to calculate the term value for n = 5substituting it in the formula we get:

The5 = 2 . (1/2)5-1 = 2 . (1/2)4 = 1/8

The resemblance between arithmetic and geometric progressions is apparently great. But we find the first substantial difference at the moment of its definition. While arithmetic progressions are formed by summing the same amount over and over again, in geometric progressions the terms are generated by multiplication, also repeated, by the same number. The differences don't stop there.

Note that when an arithmetic progression has a positive reason, that is, r> 0, each term of yours is longer than the previous one. So it is a growing progression. On the contrary, if we have a negative reason arithmetic progression, r <0, your behavior will be decreasing. Also note how quickly progression grows or slows. This is a direct consequence of the absolute value of the ratio, | r |. So the bigger it is r, in absolute value, the higher the growth rate will be and vice versa.

Next: Sum of the First n Terms of a PG