Question

The sum of an A.P. with n terms is \(n^2 + 2n + 1\) then its 6th term is

• A. 29
• B. 19
• C. 15
• D. none of these

Hint:

A quick way to find \(a_n\) from a quadratic \(S_n = An^2+Bn+C\) is to use the formula \(a_n = 2An + B-A\). Here, \(S_n = 1n^2+2n+1\), so \(A=1, B=2\), which gives \(a_n = 2(1)n + 2-1 = 2n+1\). For \(n=6\), \(a_6=2(6)+1=13\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The nth term (\(a_n\)) of a sequence can be found from the formula for the sum of its first n terms (\(S_n\)). The nth term is the difference between the sum of the first n terms and the sum of the first (n-1) terms. Note that for a sequence to be an A.P., its sum \(S_n\) must be a quadratic expression of the form \(An^2 + Bn\). The given \(S_n\) has a constant term (+1), which means the sequence generated is not a true A.P. However, we proceed by applying the standard method to find the term.

Step 2: Key Formula or Approach:
The formula to find the nth term (\(a_n\)) from the sum of n terms (\(S_n\)) is:
\[ a_n = S_n - S_{n-1} \] We will use this formula to find the 6th term, \(a_6\).

Step 3: Detailed Explanation:
The given formula for the sum of n terms is:
\[ S_n = n^2 + 2n + 1 \] First, calculate the sum of the first 6 terms (\(S_6\)):
\[ S_6 = (6)^2 + 2(6) + 1 = 36 + 12 + 1 = 49 \] Next, calculate the sum of the first 5 terms (\(S_5\)):
\[ S_5 = (5)^2 + 2(5) + 1 = 25 + 10 + 1 = 36 \] Now, find the 6th term (\(a_6\)) by subtracting \(S_5\) from \(S_6\):
\[ a_6 = S_6 - S_5 = 49 - 36 = 13 \] The calculated 6th term is 13. This value is not present in options (A), (B), or (C).

Step 4: Final Answer:
Since the calculated 6th term is 13 and this is not among the given options, the correct choice is (D) none of these.