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S_n represents the sum of the first n terms of sequence X. I

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BrentGMATPrepNow
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S_n represents the sum of the first n terms of sequence X. I [#permalink] New post   Updated on: 23 Jul 2022, 07:36
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\(S_n\) represents the sum of the first \(n\) terms of sequence X. If \(S_n = n^2 – 6n + 9\), what is the value of 13th term of sequence X?

(A) -80
(B) 100/13
(C) 19
(D) 81
(E) 100
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C

Originally posted by BrentGMATPrepNow on 21 Jul 2022, 07:45.
Last edited by BrentGMATPrepNow on 23 Jul 2022, 07:36, edited 1 time in total.
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Re: S_n represents the sum of the first n terms of sequence X. I [#permalink] New post   Updated on: 23 Jul 2022, 07:36
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BrentGMATPrepNow wrote:
\(S_n\) represents the sum of the first \(n\) terms of sequence X. If \(S_n = n^2 – 6n + 9\), what is the value of 13th term of sequence X?

(A) -80
(B) 100/13
(C) 19
(D) 81
(E) 100


Notice that \(S_13 = term_1 + term_2 + term_3 + . . . term_{12} + term_{13}\)
And \(S_{12} = term_1 + term_2 + term_3 + . . . term_{11} + term_{12}\)
So, \(S_{13}\) \(-\) \(S_{12}\) \(= term_{13} \)

\(S_{13} = 13^2 – 6(13) + 9 = 100\)
\(S_{12}= 12^2 – 6(12) + 9 = 81\)
Aside: By recognizing x² – 6x + 9 = (x-3)², I could have saved myself a little bit of time calculating each sum
So, \(term_{13} = 100 - 81 - 19\)

Answer: D

Originally posted by BrentGMATPrepNow on 22 Jul 2022, 07:01.
Last edited by BrentGMATPrepNow on 23 Jul 2022, 07:36, edited 1 time in total.
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S_n represents the sum of the first n terms of sequence X. I [#permalink] New post   Updated on: 23 Jul 2022, 08:18
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BrentGMATPrepNow wrote:
\(S_n\) represents the sum of the first \(n\) terms of sequence X. If \(S_n = x^2 – 6x + 9\), what is the value of 13th term of sequence X?

(A) -80
(B) 100/13
(C) 19
(D) 81
(E) 100


\(S_n = x^2 – 6x + 9\) = (x-3)^2
We need to find the 13th term of the sequence = S(First 13 terms) - S(First 12 terms) = (13-3)^2 - (12-3)^2 = 10^2 - 9^2 = 19

Answer - C

Originally posted by av1901 on 21 Jul 2022, 15:43.
Last edited by av1901 on 23 Jul 2022, 08:18, edited 1 time in total.
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Re: S_n represents the sum of the first n terms of sequence X. I [#permalink] New post   22 Jul 2022, 19:07
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BrentGMATPrepNow wrote:
\(S_n\) represents the sum of the first \(n\) terms of sequence X. If \(S_n = x^2 – 6x + 9\), what is the value of 13th term of sequence X?

(A) -80
(B) 100/13
(C) 19
(D) 81
(E) 100


I think the question meant to say S_n = n^2 - 6n + 9.

Notice that n^2 - 6n + 9 = (n - 3)^2. So, S_n = (n - 3)^2.

Notice also that the 13th term of the sequence is the difference between the sum of the first 13 terms of the sequence (which is S_13) and the sum of the first 12 terms of the sequence (which is S_12). So the question is to calculate the value of S_13 - S_12.

\(\Rightarrow\) S_13 - S_12

\(\Rightarrow\) (13 - 3)^2 - (12 - 3)^2

\(\Rightarrow\) 10^2 - 9^2

\(\Rightarrow\) (10 - 9)(10 + 9) = 1 * 19 = 19

Answer: C
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Re: S_n represents the sum of the first n terms of sequence X. I [#permalink] New post   23 Jul 2022, 07:37
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ScottTargetTestPrep wrote:
I think the question meant to say S_n = n^2 - 6n + 9.

Thanks for the heads up, Scott! I've edited the question.

Cheers,
Brent
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Re: S_n represents the sum of the first n terms of sequence X. I [#permalink] New post   18 Oct 2023, 20:13
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