Question

If the sum of the first $ n $ terms of an arithmetic progression is given by $ S_n = 3n^2 + 5n $, find the first term $ a $ and common difference $ d $.

• A. \( a=8, d=6 \)
• B. \( a=8, d=3 \)
• C. \( a=5, d=6 \)
• D. \( a=3, d=5 \)

Hint:

Tip: Remember, \( t_n = S_n - S_{n-1} \) is useful to find terms from sum expressions.

Answer 1

The Correct Option is A

Given the sum of the first n terms of an arithmetic progression (AP) as \( S_n = 3n^2 + 5n \), we aim to find the first term \( a \) and the common difference \( d \).

Step 1: Equate Sum Formulas

The standard formula for the sum \( S_n \) of the first \( n \) terms of an AP is:

\( S_n = \frac{n}{2}(2a + (n-1)d) \)

We set the given expression for \( S_n \) equal to the standard formula:

\( \frac{n}{2}(2a + (n-1)d) = 3n^2 + 5n \)

Multiply both sides by 2:

\( n(2a + (n-1)d) = 6n^2 + 10n \)

Divide both sides by \( n \):

\( 2a + (n-1)d = 6n + 10 \)

Step 2: Determine \( a \) and \( d \)

To find \( a \), substitute \( n = 1 \):

\( 2a + (1-1)d = 6(1) + 10 \)

\( 2a = 16 \)

\( a = 8 \)

To find \( d \), substitute \( n = 2 \) and the value of \( a \):

\( 2a + (2-1)d = 6(2) + 10 \)

\( 2(8) + d = 22 \)

\( 16 + d = 22 \)

\( d = 6 \)

Solution: The first term \( a \) is 8, and the common difference \( d \) is 6. This corresponds to the option \((a=8, d=6)\).