Question

The sum of the first \( n \) terms of an arithmetic progression is \( S_n = 3n^2 + 2n \). Find the 5th term of the sequence.

• A. 15
• B. 29
• C. 19
• D. 21
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Hint:

To find the \( n \)-th term of an arithmetic progression given the sum \( S_n \), use \( a_n = S_n - S_{n-1} \).

Answer 1

The Correct Option is B

The sum of the first \( n \) terms of an arithmetic progression is given by \( S_n = 3n^2 + 2n \). The \( n \)-th term \( a_n \) is found using: \[ a_n = S_n - S_{n-1} \] Calculate \( S_{n-1} \): \[ S_{n-1} = 3(n-1)^2 + 2(n-1) = 3(n^2 - 2n + 1) + 2n - 2 = 3n^2 - 6n + 3 + 2n - 2 = 3n^2 - 4n + 1 \] Thus: \[ a_n = S_n - S_{n-1} = (3n^2 + 2n) - (3n^2 - 4n + 1) = 3n^2 + 2n - 3n^2 + 4n - 1 = 6n - 1 \] For the 5th term (\( n = 5 \)): \[ a_5 = 6 \cdot 5 - 1 = 30 - 1 = 29 \] This does not match any option. Let’s verify by checking the first few terms: \[ S_1 = 3 \cdot 1^2 + 2 \cdot 1 = 5 \implies a_1 = 5 \] \[ S_2 = 3 \cdot 2^2 + 2 \cdot 2 = 12 + 4 = 16 \implies a_2 = 16 - 5 = 11 \] \[ S_3 = 3 \cdot 3^2 + 2 \cdot 3 = 27 + 6 = 33 \implies a_3 = 33 - 16 = 17 \] \[ S_4 = 3 \cdot 4^2 + 2 \cdot 4 = 48 + 8 = 56 \implies a_4 = 56 - 33 = 23 \] \[ S_5 = 3 \cdot 5^2 + 2 \cdot 5 = 75 + 10 = 85 \implies a_5 = 85 - 56 = 29 \] The sequence is 5, 11, 17, 23, 29. The 5th term is : \[ {29} \]