Question

If the sum of the first \(n\) terms of an A.P. is \(S_n = 3n^2 + 5n\), find the \(10^{th}\) term.

• A. \(60\)
• B. \(61\)
• C. \(62\)
• D. \(63\)

Hint:

Whenever the formula for \(S_n\) is given, use \[ T_n = S_n - S_{n-1}. \] This is the fastest way to find a specific term without expanding the entire sequence.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem provides a formula for the sum of the first \(n\) terms (\(S_n\)) of an Arithmetic Progression.
We are asked to find the specific value of the \(10^{th}\) term (\(T_{10}\)).
Step 2: Key Formula or Approach:
The \(n^{th}\) term of any sequence can be found if the sum of the first \(n\) terms is known, using the relation:
\[ T_n = S_n - S_{n-1} \] Step 3: Detailed Solution:
To find the \(10^{th}\) term, we substitute \(n=10\) into the formula:
\[ T_{10} = S_{10} - S_{9} \] First, compute \(S_{10}\) by substituting \(n=10\) into the given \(S_n\) formula:
\[ S_{10} = 3(10)^2 + 5(10) \] \[ S_{10} = 3(100) + 50 \] \[ S_{10} = 300 + 50 = 350 \] Next, compute \(S_9\) by substituting \(n=9\):
\[ S_9 = 3(9)^2 + 5(9) \] \[ S_9 = 3(81) + 45 \] \[ S_9 = 243 + 45 = 288 \] Finally, find \(T_{10}\) by subtracting \(S_9\) from \(S_{10}\):
\[ T_{10} = 350 - 288 \] \[ T_{10} = 62 \] Step 4: Final Answer:
The \(10^{th}\) term of the given Arithmetic Progression is \(62\).