Find a formula for Sn, the sum of the first n terms of the geometric sequence

Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is also known as a geometric sequence of numbers that follow a pattern.

Sum of n terms

Consider the G.P,

\(a,ar,ar^2,…..ar^{n-1}\)

Let \(S_n,a,r\) be the sum of n terms, first term and ratio of the G.P respectively.

Then, \(S_n\) = \(a + ar + ar^2 + ⋯ + ar^{n-1}\) —(1)

There are two cases, either \(r = 1\) or \(r ≠ 1\) If r=1, then

\(S_n\) = \( a + a + a + ⋯ a\) = \(na\)

When \(r ≠ 1\), Multiply (1) with r gives, \(rS_n\)

= \( ar + ar^2 + ar^3 + ⋯ + ar^{n-1} + ar^n\)—(2)

Subtracting (1) from (2) gives

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

Sum of n terms n = \(\frac{a(r^n – 1)}{r-1}\); Where r \(\neq\) 1

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