If \(S_n\) is the sum of first \(n\) terms of a certain sequence and \(S_n = n (n^2 + 1)\) for all positive integers.

[Vincen]

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

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

Answer: C

Source: e-GMAT

[Brent@GMATPrepNow]

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

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

Answer: C

Source: e-GMAT

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