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 and working with algebraic identities. A medium difficulty GMAT 650 level problem solving question.

Question 20: 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?

- 3
- \\frac {1} {3})
- 2
- 9
- \\frac {1} {9})

@ INR

Formula 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 {a\left({r}^{6}-1\right)} {r-1})

And sum of the first 3 terms of the G.P. will be equal to \\frac {a\left ( r^3-1\right)} {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 {a\left(r^6 - 1\right)} {r - 1}} {\frac {a\left(r^3 - 1\right)} {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 r^{3} + 1 = 9

r^{3} = 8

r = 2

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