Progression in Geometry A GP's general form is a, ar, ar2, ar3, and so on. Tn = arn-1, where a = initial term and r = common ratio = Tn/Tn-1, is the nth term of a GP series. S = a/(1-r) where 0 r1 is the sum of infinite terms of a GP series. If the first term is a, the common ratio of **a finite G.P.** is r. The second term can be determined from the formula S = a/(1-r).

In an infinite G.P., such as the geometric series, the formula becomes more complicated. It turns out that the value of the nth term of an infinite G.P. is equal to the product of the first n terms, divided by the nth term of the original series.

For example, if we were to find **the 10th term** of a G.P. with a = 1 and r = 0.5, we would use the formula T10 = ar10-1. In **this case**, T10 = 0.125. We can verify that 0.125 is the right answer by multiplying together **the first 10 terms** of the series: 1*0.5*0.25*0.125*0.0625*0.03125*0.015625*0.0078125*0.000984375. This number equals 0.125.

You should know that there are different forms of G.Ps. For example, there is also a rational GP where a and b are integers.

Formulas for Geometric Progression The first word is a, and the common ratio is r. The common ratio r = l/[r(n-1)] and the nth phrase from the end of the GP. The sum of infinite, that is, the sum of a GP with infinite terms, is S = a/(1-r), where 0 r 1. In mathematics, geometric progression is used to describe a sequence in which each term after the first is obtained by multiplying the previous term by some number called the "common ratio". For example, if we start with $3100$ and multiply it by $0.9$, the result is $27900$. If we continue this process, we will get $974700$, $38814000$, etc.

In physics, geometrical progression describes an expansion or contraction of something where each item is divided into **the same part** as the last (usually length) or else increases or decreases in size by **a constant percentage**. For example, if you double the size of everything every week for months at a time, you are using a geometrical progression. Double your money every year and you'll have **twice as much money** when you retire!

As another example, the diameter of a circle increases by a constant percentage each time it is doubled. This is because the radius of a circle is equal to its diameter multiplied by $1.41$. So if the diameter increases by 10 percent, then the radius must decrease by 10 percent too, or else the circle would not be as big.

Tn = arn-1 is the nth term of a GP. The usual ratio is r = Tn/Tn-1. Sn = a [(rn-1)/(r-1)] for determining the sum of the first n terms of a GP if r 1 and r > 1. Also, sn = an [r/(r+1)] for **arbitrary integers** r > 0.

A geometric progression is a succession of phrases in which each following term is formed by multiplying the prior term by a constant number. (GP), whereas the constant number is referred to as the common ratio. For instance, 2, 4, 8, 16, 32, 64,... is a GP with a common ratio of 2. The first few terms of the GP are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024..