Geometric Progression (GP)

In the lesson, we'll learn about geometric progression.

We'll cover the following

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

Here the first term, aa, is 33, a=3a=3, and the common ratio, rr, is 22, r=2r=2.

In general,

a,ar,ar2,ar3,...,arn1a, ar, ar^2, ar^3, ... , ar^{n-1}

Where the nthnth term =arn1= ar^{n-1}

Sum

The sum of GP with nn terms is

a+ar+ar2+...+arn1a + ar + ar^2 + ... + ar^{n-1}

=(1r)(a+ar+ar2+...+arn1)1r= \frac{(1-r)(a + ar + ar^2 + ... + ar^{n-1})}{1-r}

=(a+ar+ar2+...+arn1)(ar+ar2+ar3+...+arn)1r= \frac{(a + ar + ar^2 + ... + ar^{n-1}) - (ar + ar^2 + ar^3 + ... + ar^{n})}{1-r}

=aarn1r= \frac{a-ar^{n}}{1-r}

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


Infinite geometric progression

This is a special and very useful case of GP when the common ratio is r<1r < 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 nn \to \infty

Sum

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

We have

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

As n,rn0n \to\infty, r^n \to 0

sum=a1rsum = \frac{a}{1-r}


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

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