Question: If Sn is the sum of the first n terms of a certain sequence and if Sn = n(n + 1) for all positive integers n, what is the third term of the sequence?

A. 3
B. 4
C. 6
D. 8
E. 9

Answer: C

Approach Solution (1):
Let there be 3 terms in the sequence, namely a, b & c
Given that a + b + c = 3 * 4 = 12………….. (1)
Now say there are 2 terms, namely a & b
So, a + b = 2 * 3 = 6………. (2)
Substitute value of a+b from eqn (2) in eqn (1)
6 + c = 12
c = 6
Correct option: C

Approach Solution (2):
For the given condition, Sn=n(n+1) where n=positive integers
We start substituting with n=1 –
S1 = 1(1+1) = 2
S2 = 6
S3 = 12
Now, for n=1, Sn=2, thus sequence starts with number 2.
Then, sum of first two nos is S2 which is 6. Thus, first two nos are 2,4
Next, sum of first three nos is S3=12 i.e. first 3 numbers are 2,4,6
From the above sequence, the 3rd number is 6
Correct option: C

Approach Solution (3):
Use this formula when Sn is given
Tn = nth term of term of the series
Tn = Sn - S(n-1)
and T1 = S1
T1 = 2
T2 = S2 - S1 = 6 – 2 = 4
T3 = S3 - S2 = 12 – 6 = 6
Correct option: C

“If Sn is the sum of the first n terms of a certain sequence and if Sn = n(n + 1) for all positive integers n, what is the third term of the sequence?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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