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

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New post 14 Feb 2016, 05:38
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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
[Reveal] Spoiler: OA

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