Question

The sum of first n terms of an A.P. is $S_n = n^2 + 4n$. Find the 10th term.

• A. 32
• B. 23
• C. 46
• D. 18

Hint:

For any quadratic expression for S\(_n\) of the form \(An^2 + Bn\), the n\(^{th}\) term is given by \(a_n = S_n - S_{n-1}\). Alternatively, the common difference 'd' is \(2A\), and the first term \(a_1\) is \(A+B\). Here, \(A=1, B=4\), so \(d=2(1)=2\) and \(a_1=1+4=5\). The 10th term is \(a_{10} = a_1 + (10-1)d = 5 + 9(2) = 5+18=23\).

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 of an Arithmetic Progression (A.P.), which is \(S_n = n^2 + 4n\). We need to find the value of 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 by taking the difference between the sum of the first 'n' terms and the sum of the first '(n-1)' terms.
The formula is: \[ a_n = S_n - S_{n-1} \] To find the 10th term, we will use \(n = 10\).
\[ a_{10} = S_{10} - S_{9} \] Step 3: Detailed Explanation:
First, we calculate the sum of the first 10 terms (\(S_{10}\)) using the given formula.
\[ S_{10} = (10)^2 + 4(10) \] \[ S_{10} = 100 + 40 = 140 \] Next, we calculate the sum of the first 9 terms (\(S_{9}\)).
\[ S_{9} = (9)^2 + 4(9) \] \[ S_{9} = 81 + 36 = 117 \] Now, we can find the 10th term by substituting these values into the formula from Step 2.
\[ a_{10} = S_{10} - S_{9} \] \[ a_{10} = 140 - 117 = 23 \] Step 4: Final Answer:
The 10th term of the A.P. is 23.