This post is about the sum of a finite Geometric Progression. For the sum of an infinite Geometric Progression, go to this post.

The formula

Let 'a' be the first term, 'r' be the common ratio and 'n' be the number of terms in a finite G.P. Then, sum of the 'n' terms of the G.P. depends on the value of 'r'.

If 'r' is between -1 and 1 (i.e., |r| is smaller than 1)

Let 'a' be the first term and 'r' be the common ratio of a Geometric Progression. Then the Geometric Progression can be represented by

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The formula

Let 'a' be the first term, 'r' be the common ratio and 'n' be the number of terms in a finite G.P. Then, sum of the 'n' terms of the G.P. depends on the value of 'r'.

If 'r' is between -1 and 1 (i.e., |r| is smaller than 1)

Sn = a(1 - r^n)/(1 - r)If 'r' is greater than 1 or smaller than or equal to -1 (i.e., |r| > 1 or r = -1)

Sn = a(r^n - 1)/(r - 1)If 'r' is equal to 1 (i.e., the G.P. is constant),

Sn = naDerivation of the formula

Let 'a' be the first term and 'r' be the common ratio of a Geometric Progression. Then the Geometric Progression can be represented by

a, ar, ar^2, ar^3, ... upto ar^(n-1)Sum of the Geometric Progression is

Sn = a + ar + ar^3 + ... + ar^(n - 1) ............ (1)Multiplying 'r' to both sides,

r . Sn = ar + ar^2 + ar^4 + ... + ar^n ................ (2)Subtract equation (2) from equation (1),

Sn - r . Sn = a - ar^nSimplify further,

Sn(1 - r) = a(1 - r^n)

Sn = a(1 - r^n)/(1 - r)This formula is used when the common ratio of the G.P. is lesser than 1 or greater than -1, that is, it is in between 1 and -1. When the common ratio is greater than 1, or lesser than or equal to -1 (i.e., r <= -1 and r > 1) then this formula is altered to

Sn = a(r^n - 1)/(r - 1)On the other hand, when the common ratio is equal to 1, the G.P. is a constant G.P., that is, all terms of the G.P. are equal to the first term. Then, the sum of its 'n' terms is given by

Sn = na

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