Question

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

• A. \(60\)
• B. \(62\)
• C. \(64\)
• D. \(66\)

Hint:

Whenever the sum of the first \(n\) terms \(S_n\) is given, the \(n^{th}\) term can be quickly obtained using \(a_n = S_n - S_{n-1}\). This is one of the fastest methods to extract individual terms from a sum formula.

Answer 1

The Correct Option is B

Topic: Sequence and Series (Arithmetic Progression)
Step 1: Understanding the Question:
The problem provides the formula for the sum of the first \(n\) terms (\(S_n\)) of an Arithmetic Progression.
We are required to find a specific term, the \(10^{th}\) term (\(a_{10}\)).
Step 2: Key Formula or Approach:
The relationship between the \(n^{th}\) term and the sum of terms is given by:
\[ a_n = S_n - S_{n-1} \] To find the \(10^{th}\) term, we calculate the difference between the sum of the first 10 terms and the sum of the first 9 terms.
Step 3: Detailed Explanation:
First, we calculate \(S_{10}\) by substituting \(n = 10\) into the given formula:
\[ S_{10} = 3(10)^2 + 5(10) \] \[ S_{10} = 3(100) + 50 = 300 + 50 = 350 \]
Next, we calculate \(S_9\) by substituting \(n = 9\):
\[ S_9 = 3(9)^2 + 5(9) \] \[ S_9 = 3(81) + 45 = 243 + 45 = 288 \]
Now, we find the \(10^{th}\) term using the subtraction rule:
\[ a_{10} = S_{10} - S_9 \] \[ a_{10} = 350 - 288 = 62 \]
Step 4: Final Answer:
The value of the \(10^{th}\) term is 62.