Question

If the sum of \( n \) terms of an A.P. is \( 3n^2 + 5n \) and its \( m \)-th term is 164, then the value of \( m \) is:

• A. \( 26 \)
• B. \( 27 \)
• C. \( 28 \)
• D. \( 29 \)

Hint:

If sum is given: \begin{itemize} \item Use \( a_n = S_n - S_{n-1} \). \item This converts quadratic sums into linear terms. \end{itemize}

Answer 1

The Correct Option is B

Concept: Nth term from sum formula: \[ a_n = S_n - S_{n-1} \] Step 1: {\color{red}Compute general term.} \[ S_n = 3n^2 + 5n \] \[ S_{n-1} = 3(n-1)^2 + 5(n-1) \] \[ = 3n^2 - 6n + 3 + 5n - 5 \] \[ = 3n^2 - n - 2 \] So: \[ a_n = (3n^2 + 5n) - (3n^2 - n - 2) \] \[ = 6n + 2 \] Step 2: {\color{red}Use given term.} \[ a_m = 164 \] \[ 6m + 2 = 164 \] \[ 6m = 162 \Rightarrow m = 27 \]