Question

Choose the correct answer in each of the following questions:

The sum of first n terms of an AP is (5n − n²). The nth term of the AP is

(a) (5 − 2n) (b) (6 − 2n) (c) (2n − 5) (d) (2n − 6)

Answer 1

Let S_n denotes the sum of first n terms of the AP.
$$\begin{gathered} \therefore S_n=5n-n^2 \\ \Rightarrow S_{n-1}=5\left(n-1\right)-{\left(n-1\right)}^2 \\ =5n-5-n^2+2n-1 \\ =7n-n^2-6 \end{gathered}$$

∴ n^th term of the AP, a_n = S_n − S_{n − 1
$$\begin{gathered} =\left(5n-n^2\right)-\left(7n-n^2-6\right) \\ =6-2n \end{gathered}$$}

Thus, the n^th term of the AP is (6 − 2n).

Hence, the correct answer is option B.