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The sum of the terms of a geometric progression is 2047.

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The sum of the terms of a geometric progression is 2047.  [#permalink]

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New post 11 Aug 2017, 08:17
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The sum of the terms of a geometric progression is 2047. Find the common ratio.

(1) The first and last terms of the series are 1 and 1024 respectively.

(2) Last but one term of the series is 512.

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Re: The sum of the terms of a geometric progression is 2047.  [#permalink]

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New post 11 Aug 2017, 12:45
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Gnpth wrote:
The sum of the terms of a geometric progression is 2047. Find the common ratio.

(1) The first and last terms of the series are 1 and 1024 respectively.

(2) Last but one term of the series is 512.


In case there are people reading this question and worrying that they haven't learned about geometric progressions and common ratios, you need not worry. The GMAT does not expect you to be familiar with these terms. Likewise, you don't need to know the formula for the sum of a geometric series/sequence/progression.

Cheers,
Brent
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The sum of the terms of a geometric progression is 2047.  [#permalink]

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New post 07 Feb 2018, 03:17
Gnpth wrote:
The sum of the terms of a geometric progression is 2047. Find the common ratio.

(1) The first and last terms of the series are 1 and 1024 respectively.

(2) Last but one term of the series is 512.




Summation of n terms of a G.P = sum = a\((\frac{ 1- r^n}{1-r})\)

Where a is the first term and n is the number of terms whose sum is being taken and r is the common ratio.



2047 = \(1 (\frac{ 1- r^n}{1-r})\)

Also nth term of a G.P.= \(ar^{(n-1)}\) where a is the first term and r is the common ratio.
So last term ( nth term) = 1024 = \(ar^{(n-1)}\)
We know a (first term) is 1
\(1024=r^{(n-1)}\)
\(2^{10}\) = 1024 here n-1= 10 so n=11
\(4^5\) = 1024 here n-1= 5 so n=6
\(32^2\) = 1024 here n-1= 2 so n=3

Out of these only n=11 will give sum of 2047 hence r= 2 is the required common ratio.

Hope this helps !
Kudos will be appreciated !
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Re: The sum of the terms of a geometric progression is 2047.  [#permalink]

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New post 08 Jun 2019, 12:14
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As Brent says above, you don't need to know anything about "geometric series" for the GMAT - you don't even need to know what they are. So ignoring the other problems with the wording of the question, it's simply out of the scope of the test. The only way it could appear is if the question itself also provided the formulas you'd need to use to answer it.

But since the only solution above assumes r is a positive integer, and uses inspection, I can offer a different method, though test takers can safely ignore questions like this. We know from Statement 1 that the first term a = 1, and the last term is 1024. In a geometric sequence with n terms, the last term is equal to (a)(r^(n-1)). Since a =1, we know

r^(n-1) = 1024

The sum S of a geometric sequence with n terms is equal to

S = a(1- r^n)/(1 - r)

so since that sum is 2047, we also know, using that a=1,

2047 = (1 - r^n) / (1 - r)

Since r^n is just equal to (r)(r^(n-1)) by basic exponent rules, and since r^(n-1) = 1024, we can replace r^n with 1024r. So

2047 = (1 - 1024r) / (1 - r)
2047 - 2047r = 1 - 1024r
2046 = 1023r
2 = r

and Statement 1 is sufficient. Statement 2 is not sufficient (it could be the sequence with r=2 that we found in Statement 1, but since r need not be an integer, it could just be a two-term sequence where the first term is 512, and the second is 1535, among many possibilities), so the answer is A.
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Re: The sum of the terms of a geometric progression is 2047.  [#permalink]

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New post 06 Jun 2020, 22:02
BrentGMATPrepNow wrote:
Gnpth wrote:
The sum of the terms of a geometric progression is 2047. Find the common ratio.

(1) The first and last terms of the series are 1 and 1024 respectively.

(2) Last but one term of the series is 512.


In case there are people reading this question and worrying that they haven't learned about geometric progressions and common ratios, you need not worry. The GMAT does not expect you to be familiar with these terms. Likewise, you don't need to know the formula for the sum of a geometric series/sequence/progression.

Cheers,
Brent


Thank you Brent for Clarification!
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Re: The sum of the terms of a geometric progression is 2047.   [#permalink] 06 Jun 2020, 22:02

The sum of the terms of a geometric progression is 2047.

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