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

Concept: If \(S_n\) represents the sum of the first \(n\) terms of a sequence, then the \(n^{th}\) term can be obtained using the relation: \[ a_n = S_n - S_{n-1} \] This formula helps us determine a specific term by subtracting the sum of the first \(n-1\) terms from the sum of the first \(n\) terms.

Step 1: Find the value of \(S_{10}\). Given: \[ S_n = 3n^2 + 5n \] Substituting \(n = 10\): \[ S_{10} = 3(10)^2 + 5(10) \] \[ S_{10} = 3(100) + 50 \] \[ S_{10} = 300 + 50 = 350 \]

Step 2: Find the value of \(S_9\). Substituting \(n = 9\): \[ S_9 = 3(9)^2 + 5(9) \] \[ S_9 = 3(81) + 45 \] \[ S_9 = 243 + 45 = 288 \]

Step 3: Find the \(10^{th}\) term. \[ a_{10} = S_{10} - S_9 \] \[ a_{10} = 350 - 288 \] \[ a_{10} = 62 \]