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The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ……. If

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The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ……. If  [#permalink]

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New post 25 Sep 2018, 03:42
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E

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[Math Revolution GMAT math practice question]

The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ……. If n is a positive integer, what could be the value of Sn-Sn-1 in terms of n?

\(A. n\)
\(B. n^2\)
\(C. –n\)
\(D. -n^2\)
\(E. (n-1)^2\)

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Re: The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ……. If  [#permalink]

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New post 25 Sep 2018, 04:09
We see that the increment in the given sequence is:
From S1 to S2: +1 = 1^2
From S2 to S3: +4 = 2^2
From S3 to S4: +9 = 3^2
From S4 to S5: +16 = 4^2
...
Thus from Sn-1 to Sn = (n-1)^2.

Answer E
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Re: The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ……. If  [#permalink]

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New post 25 Sep 2018, 05:30
MathRevolution wrote:
[Math Revolution GMAT math practice question]

The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ……. If n is a positive integer, what could be the value of Sn-Sn-1 in terms of n?

\(A. n\)
\(B. n^2\)
\(C. –n\)
\(D. -n^2\)
\(E. (n-1)^2\)


To answer any question of sequences, one needs to find a pattern so we keep looking at what we get when we canculate expressions like \(S_n - S_{n-1}\)

\(S_1 = 1\)
\(S_2 = 2\) i.e. \(S_2 - S_{1} = 1= 1^1\)
\(S_3 = 6\) i.e. \(S_3 - S_{2} = 4 = 2^2\)
\(S_4 = 15\) i.e. \(S_4 - S_{3} = 9 = 3^2\)

i.e. \(S_n - S_{n-1} = (n-1)^2\)

Answer: Option E
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Re: The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ……. If  [#permalink]

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New post 25 Sep 2018, 09:57
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Top Contributor
MathRevolution wrote:
[Math Revolution GMAT math practice question]

The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ……. If n is a positive integer, what could be the value of Sn-Sn-1 in terms of n?

\(A. n\)
\(B. n^2\)
\(C. –n\)
\(D. -n^2\)
\(E. (n-1)^2\)


If you don't spot a pattern, you can always TEST the answer choices

We want to find Sn - Sn-1
So, let's test a value of n.
How about n = 2
So, we want to find the value of S2 - S1
Well, we're told that S2 = 2 and S1 = 1
So, S2 - S1 = 2 - 1 = 1

So, when n = 2, the answer to the question (what is the value of Sn - Sn-1) is 1

Now we'll check the answer choices to see which one yields an OUTPUT of 1 when we input n = 2
A. n = 2. NO GOOD. We need an output of 1. ELIMINATE
B. n² = 2² = 4. NO GOOD. We need an output of 1. ELIMINATE
C. -n = -2 = -2. NO GOOD. We need an output of 1. ELIMINATE
D. -n² = -2² = -4. NO GOOD. We need an output of 1. ELIMINATE
E. (n - 1)² = (2 - 1)² = 1. perfect!!

Answer: E

Cheers,
Brent
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Re: The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ……. If  [#permalink]

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New post 27 Sep 2018, 00:56
=>

We begin by writing out the terms of the sequence:

S2-S1 \(= 2 -1 = 1=1^2\)
S3-S2 \(= 6 – 2 = 4 = 2^2\)
S4-S3 \(= 15 – 6 = 9 = 3^2\)
S5-S4 \(= 31-15 = 16 = 4^2\)

Following the above pattern gives
Sn-Sn-1 \(= (n-1)^2\)

Therefore, the answer is E.
Answer: E
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Re: The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ……. If &nbs [#permalink] 27 Sep 2018, 00:56
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