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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