The sum of the first k terms of a series S is $3k{}^2+5k$. Which one of the following is correct?

Given:

The sum of the first k terms of a series S is given as S_k = 3k² + 5k.

Concept:

To find the nth term of a series, we use the formula:

a_n = S_n - S_n-1

If the nth term forms an arithmetic progression (AP), the common difference (d) is given by:

d = a_n+1 - a_n

Calculation:

We are given S_k = 3k² + 5k.

⇒ a_n = S_n - S_n-1

Also S_n = 3n² + 5n and S_n-1 = 3(n-1)² + 5(n-1)

⇒ a_n = [3n² + 5n] - [3(n-1)² + 5(n-1)]

⇒ a_n = [3n² + 5n] - [3(n² - 2n + 1) + 5n - 5]

⇒ a_n = [3n² + 5n] - [3n² - 6n + 3 + 5n - 5]

⇒ a_n = 3n² + 5n - 3n² + 6n - 3 - 5n + 5

⇒ a_n = 6n + 2

The series is an arithmetic progression (AP) if the difference between consecutive terms is constant.

⇒ Common difference d = a_n+1 - a_n

⇒ a_n+1 = 6(n+1) + 2 = 6n + 6 + 2 = 6n + 8

⇒ d = (6n + 8) - (6n + 2) = 6

∴ The terms of the series form an arithmetic progression with a common difference of 6.

Hence, the correct answer is Option B.