If Sn is the sum of first n terms of a certain sequence and Sn

Question

If Sn is the sum of first n terms of a certain sequence and Sn = n (n^2 + 1) for all positive integers. What is the 4th term of the sequence?

A. 10
B. 20
C. 38
D. 66
E. 68

BrentGMATPrepNow · Accepted Answer

GIVEN: Sn is the sum of first n terms of a certain sequence  
So, for example, 
 S4 = term1 + term2 + term3 + term4 
 S3 = term1 + term2 + term3

This means that  S4  -  S3  = ( term1 + term2 + term3 + term4 ) - ( term1 + term2 + term3  )
=  term4  
Perfect, we now know that  term4  =  S4  -  S3

According to the formula for Sn, we know that:
 S4 = 4(4² + 1) = 4(16 + 1) = 68 
 S3 = 3(3² + 1) = 3(9 + 1) = 30

We can now write:  term4  =  68  -  30  = 38

Answer: C

Cheers,
Brent

Condonsation · Answer

GMATPrepNow  Thanks for the explanation! Don't we have to know that all the terms are positive before we can assume S4 - S3 = (term1 + term2 + term3 + term4) - (term1 + term2 + term3)?

BrentGMATPrepNow · Answer

It doesn't matter whether the terms are positive or negative. 
If S4 = (term1 + term2 + term3 + term4) -
And S3 = (term1 + term2 + term3)
Then S4 - S3 = (term1 + term2 + term3 + term4) - (term1 + term2 + term3)
= term1 + term2 + term3 + term4 - term1 - term2 - term3
= term1 - term1 + term2 - term2 + term3 - term3 + term4
= 0 + 0 + 0 + term4
= term4

Cheers, 
Brent

ScottTargetTestPrep · Answer

Solution:

If we let a1, a2, a3, … be the terms of the sequence, we should recognize that S1 = a1, S2 = a1 + a2, S3 = a1 + a2 + a3, and so on. Therefore, using the formula given for Sn, we have:

S1 = 1(1^2 + 1) = 1(2) = 2 = a1 → a1 = 2

S2 = 2(2^2 + 1) = 2(5) = 10 = a1 + a2 → 10 = 2 + a2 → a2 = 8

S3 = 3(3^2 + 1) = 3(10) = 30 = a1 + a2 + a3 → 30 = 2 + 8 + a3 → a3 = 20

S4 = 4(4^2 + 1) = 4(17) = 68 = a1 + a2 + a3 + a4 → 68 = 2 + 8 + 20 + a4 → a4 = 38

Alternate Solution:

If we let a_n be the nth term of the sequence, we should recognize that a_1 = S_1 and for n > 1, a_n = S_n - S_(n - 1).

Since we are looking for a_4, using the formula given for S_n, we have:

a_4 = S_4 - S_3

a_4 = 4(4^2 + 1) - 3(3^2 + 1)

a_4 = 4(17) - 3(10) = 68 - 30 = 38

Answer: C

Crytiocanalyst · Answer

t-4 = S-4 - S-3

t-4 = 4(4^2 + 1) - 3(3^2 + 1)

a-4 = 4(17) - 3(10) = 68 - 30 = 38

Hence IMO C

peaky · Answer

I found this question very confusing, how to identify questions like this?

If Sn is the sum of first n terms of a certain sequence and Sn

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