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

Concept: If \(S_n\) denotes the sum of the first \(n\) terms of a sequence, then the \(n^{th}\) term is given by \[ T_n = S_n - S_{n-1} \] This relation helps us find an individual term when the sum formula is given.

Step 1: Use the relation between \(T_n\) and \(S_n\). \[ T_n = S_n - S_{n-1} \] For \(n=10\), \[ T_{10} = S_{10} - S_{9} \]

Step 2: Compute \(S_{10}\). \[ S_n = 3n^2 + 5n \] \[ S_{10} = 3(10)^2 + 5(10) \] \[ S_{10} = 300 + 50 \] \[ S_{10} = 350 \]

Step 3: Compute \(S_9\). \[ S_9 = 3(9)^2 + 5(9) \] \[ S_9 = 3(81) + 45 \] \[ S_9 = 243 + 45 \] \[ S_9 = 288 \]

Step 4: Find the \(10^{th}\) term. \[ T_{10} = S_{10} - S_9 \] \[ T_{10} = 350 - 288 \] \[ T_{10} = 62 \]