Question

The sum of first n terms of an A.P. is S\(_n\) = n\(^2\) + 4n. Find the 10\(^{th}\) term.

• A. 32
• B. 23
• C. 46
• D. 18

Hint:

When S\(_n\) is a quadratic expression in 'n' (like An\(^2\) + Bn), the sequence is always an A.P. The n\(^{th}\) term can also be found by taking the derivative with respect to n and adjusting the constant: a\(_n\) = 2An + (B-A). In this case, a\(_n\) = 2(1)n + (4-1) = 2n + 3. So, a\(_{10}\) = 2(10) + 3 = 23.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given the formula for the sum of the first 'n' terms (S\(_n\)) of an Arithmetic Progression (A.P.). We need to find the 10th term (a\(_{10}\)) of this A.P.
Step 2: Key Formula or Approach:
The n\(^{th}\) term of an A.P. can be found from the sum of n terms using the relation: \[ a_n = S_n - S_{n-1} \] This formula works because the sum up to n terms minus the sum up to (n-1) terms leaves only the n\(^{th}\) term.
Step 3: Detailed Explanation:
To find the 10th term (a\(_{10}\)), we will use the formula with n = 10: \[ a_{10} = S_{10} - S_{9} \] First, let's calculate S\(_{10}\) using the given formula S\(_n\) = n\(^2\) + 4n: \[ S_{10} = (10)^2 + 4(10) = 100 + 40 = 140 \] Next, let's calculate S\(_{9}\): \[ S_{9} = (9)^2 + 4(9) = 81 + 36 = 117 \] Now, we can find a\(_{10}\): \[ a_{10} = 140 - 117 = 23 \] Step 4: Final Answer:
The 10th term of the A.P. is 23. Therefore, option (B) is the correct answer.