Question

Very-Short and Short-Answer Questions

If the sum of first n terms is (3n² + 5n), find its common difference.

Answer 1

Let S_n denotes the sum of first n terms of the AP.
$$\begin{gathered} \therefore S_n=3n^2+5n \\ \Rightarrow S_{n-1}=3{\left(n-1\right)}^2+5\left(n-1\right) \\ =3\left(n^2-2n+1\right)+5\left(n-1\right) \\ =3n^2-n-2 \end{gathered}$$
Now,
n^th term of the AP, a_n = S_n − S_{n − 1
$$\begin{gathered} =\left(3n^2+5n\right)-\left(3n^2-n-2\right) \\ =6n+2 \end{gathered}$$}
Let d be the common difference of the AP.
$$\begin{gathered} \therefore d=a_n-a_{n-1} \\ =\left(6n+2\right)-\left[6\left(n-1\right)+2\right] \\ =6n+2-6\left(n-1\right)-2 \\ =6 \end{gathered}$$

Hence, the common difference of the AP is 6.