## Geometric progression – Introduction:

In an A.P, the difference between \(n^{th}\)

An Arithmetic Progression is in the form,

\(a, a+d, a+2d, a+3d ….. a + (n-1)d\)

Now, what if the ratio of \(n^{th}\)

For example, consider the following sequence,

\(2, 4, 8, 16, 32,……..\)

You can see that,

\(\frac{4}{2} = \frac{8}{4} = \frac{16}{8} = \frac{32}{16} = 2\)

Similarly,

Cosider a series \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16},……….\)

\(\frac{\frac{1}{2}}{1}\)

In the given examples, the ratio is a constant. Such sequences are called Geometric Progressions. It is abbreviated as G.P.

A sequence \(a_1,a_2,a_3,…….a_n,….\)

Where r is a constant which is known as common ratio and none of the terms in the sequence is zero.

## General term of a Geometric Progression:

We had learned to find the \(n^{th}\)

\(a_n \)

Similarly, in case of G.P,

Let a be the first term and r be the common ratio,

Then the second term, \(a_2 = a \times r = ar \)

Third term, \(a_{3} = a_{2} \times r = ar \times r = ar^{2}\)

Similarly, \(n^{th}\)

The terms of a finite G.P can be written as \(a, ar, ar^2, ar^3,……ar^{n-1}\)

Terms of an infinite G.P can be written as \(a, ar, ar^{2}, ar^{3}, ……ar^{n-1},…….\)

\(a + ar + ar^2 + ar^3 + ⋯ + ar^n\)

\( a + ar + ar^2 + ar^3 + ⋯ + ar^n + ⋯ \)

Example: If \(n^{th}\)

Solution: First, we have to find the common ratio

\(r\)

Since the first term, \(a\)

\(a_n\)

\(192\)

\(2^{n-1} = \frac{192}{3} = 64 = 2^6\)

\(n – 1 = 6 \)

n = 7

Therefore, 192 is \(7^{th}\)

Example: \(5^{th}\)

Solution: \(ar^4\)

\(ar^2\)

Dividing (1) by (2) gives,

\(\frac{ar^4}{ar^2}\)

\(r^2\)

\(r\)

Substituting \(r\)

\(a×4^2\)

\(a_8\)

=\( 1×4^7\)

Sum of n terms of a G.P:

Consider the G.P,

\(a,ar,ar^2,…..ar^{n-1}\)

Let \(S_n,a,r\)

Then, \(S_n\)

There are two cases, either \(r = 1\)

If r=1, then \(S_n\)

When \(r ≠ 1\)

Multiply (1) with r gives,

\(rS_n\)

Subtracting (1) from (2) gives

\(rS_n – S_n = (ar + ar^{2} + ar^{3} + …. + ar^{n-2} + ar^{n-1} + ar^{n}) – (a + ar + ar^{2} + …. + ar^{n-2} + ar^{n-1})\)

\((r – 1) S_n = ar^{n} – a = a(r^{n}-1)\)

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

Example: Find the sum of 6 terms of the G.P 4, 12, 36,…..

Solution: \(a\)

Common ratio,\(r = \frac{12}{4} = 3\)

\(n\)

Sum of n terms of a G.P,

\(S_n\)

\(S_6\)

=\(\frac{4(729-1)}{(2)}\)

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