Sum of 'n' terms of a Geometric Progression

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)

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 = na 

Derivation 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^n

Simplify 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

Related posts:

• Geometric progressions 
• Sum of infinite geometric progression 
• Progressions in general