Geometric progression is a sequence of numbers such that each term after the first is obtained by multiplying the previous one by a fixed non-zero number, called the common ratio. For example:

$3, 6, 12, 24, ...$

Here the first term, $a$, is $3$, $a=3$, and the common ratio, $r$, is $2$, $r=2$.

In general,

$a, ar, ar^2, ar^3, ... , ar^{n-1}$

Where the $nth$ term $= ar^{n-1}$

Sum

The sum of GP with $n$ terms is

$a + ar + ar^2 + ... + ar^{n-1}$

$= \frac{(1-r)(a + ar + ar^2 + ... + ar^{n-1})}{1-r}$

$= \frac{(a + ar + ar^2 + ... + ar^{n-1}) - (ar + ar^2 + ar^3 + ... + ar^{n})}{1-r}$

$= \frac{a-ar^{n}}{1-r}$

$= \frac{a(1-r^n)}{1-r}$

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Infinite geometric progression

This is a special and very useful case of GP when the common ratio is $r < 1$.

It is easy to see that this is only where terms become smaller and smaller and hence the sum converge to a value when the number of terms $n \to \infty$

Sum

The sum is easy to calculate using the formula for the sum of GP terms.

We have

$sum =\frac{a(1-r^n)}{1-r}$

As $n \to\infty, r^n \to 0$

$sum = \frac{a}{1-r}$

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In the next lesson, we’ll start learning PnC (Permutations and Combinations), starting with permutation.