# Geometric Progression

Concept: Sum of first 'n' terms of a geometric progression, Common ratio of a geometric progression.

The GMAT Math practice question given below is a sequences and series question. Concept covered is sum of a Geometric Progression and its common ratio.

#### Question: If the ratio of the sum of the first 6 terms of a G.P. to the sum of the first 3 terms of the G.P. is 9, what is the common ratio of the G.P?

1. 3
2. $$frac {1} {3}$ 3. 2 4. 9 5. $\frac {1} {9}$ #### Explanatory Answer Video explanation will be added soon. ##### Forumula to find the sum of first 'n' terms of a GP The sum of the first n terms of a G.P. is given by $\frac {{{ar}^{n}}-a} {r-1}$, where 'a' is the first term of the G.P., 'r' is the common ratio and 'n' is the number of terms in the G.P. Therefore, the sum of the first 6 terms of the G.P will be equal to $\frac {{{ar}^{6}}-1} {r-1}$ And sum of the first 3 terms of the G.P. will be equal to $\frac {{{ar}^{3}}-1} {r-1}$ ##### Use the ratio between these two sums to find 'r' The ratio of the sum of the first 6 terms : sum of first 3 terms = 9 : 1 i.e. $\frac {\frac {{{ar}^{6}}-1} {r-1}} {\frac {{{ar}^{3}}-1} {r-1}}$ = 9 $\frac {{{r}^{6}}-1} {{{r}^{3}}-1}$ = $\frac {\left$ {{{r}^{3}}+1} $right$$left$ {{{r}^{3}}-1} \right$} {{{r}^{3}}-1}) = 9

Or r3 + 1 = 9
r3 = 8
r = 2

Choice C is the correct answer.

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