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# If the ratio of the sum of the first 6 terms of a G.P. to the sum of

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Math Expert
Joined: 02 Sep 2009
Posts: 49251
If the ratio of the sum of the first 6 terms of a G.P. to the sum of  [#permalink]

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14 Feb 2016, 05:38
00:00

Difficulty:

55% (hard)

Question Stats:

70% (01:42) correct 30% (01:38) wrong based on 61 sessions

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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?

A. 3
B. 1/3
C. 2
D. 9
E. 1/9

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Joined: 02 Aug 2009
Posts: 6787
Re: If the ratio of the sum of the first 6 terms of a G.P. to the sum of  [#permalink]

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14 Feb 2016, 21:00
Bunuel wrote:
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?

A. 3
B. 1/3
C. 2
D. 9
E. 1/9

Hi,

1) formula way

the sum of n terms in a GP = first term *(1-r^n)/(1-r)..
so sum of 6 terms = first term *(1-r^6)/(1-r)..
and first three terms=first term *(1-r^3)/(1-r)..

therefore the ratio = (1-r^6)/(1-r^3)= (1-r^3)(1+r^3)/(1-r^3)= 1+r^3...
but this is given as 9..
so 1+r^3=9..
r^3=8..
or r=2..

substitution

if we have problem in simplifying after (1-r^6)/(1-r^3), substitute the choices and see when you get 9..
A. 3....
B. 1/3
C. 2
D. 9
E. 1/9
we can eliminate the fractions straightway here..B and E out..
the ratio is 9, so we cannot have r as a multiple of 9 or 3 as the numerator (1-r^6) will become [b]6th power of multiple of 3 - 1,
which will never be a multile of 9..

only C is left[/b]
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Re: If the ratio of the sum of the first 6 terms of a G.P. to the sum of  [#permalink]

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14 Feb 2016, 21:19
sum of 6 terms in GP: 1 + $$a$$ + $$a^2$$ + $$a^3$$+ $$a^4$$+ $$a^5$$ == 1 + a + $$a^2$$ + $$a^3$$ * (1 + $$a$$ + $$a^2$$ )
sum of first 3 terms in GP: 1 + $$a$$ + $$a^2$$

x= 1 + $$a$$+ $$a^2$$
given:
(x + $$a^3$$*x)/x = 9

$$a^3$$ = 8
a= 2
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Posts: 20
Re: If the ratio of the sum of the first 6 terms of a G.P. to the sum of  [#permalink]

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17 Feb 2016, 14:13
9=(a1+a2+a3+a4+a5+a6)/(a1+a2+a3)
Factorize the same terms
9=1+(a4+a5+a6)/(a1+a2+a3)
Write every term with respect to r
a1=a1
a2=a1*r^1
a3=a1*r^2
.........
9=1+(a1(r^3+r^4+r^5))/(a1(1+r^1+r^2))
8=(r^3 (1+r^1+r^2))/((1+r^1+r^2))
8=r^3
r=2
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Re: If the ratio of the sum of the first 6 terms of a G.P. to the sum of  [#permalink]

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07 Oct 2017, 00:53
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If the ratio of the sum of the first 6 terms of a G.P. to the sum of &nbs [#permalink] 07 Oct 2017, 00:53
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