Question

Find the sum of first n terms of an AP whose nth term is (5 − 6n). Hence, find the sum of its first 20 terms.

Answer 1

Let a_n be the n^th term of the AP.

∴ a_n = 5 − 6n

Putting n = 1, we get

First term, a = a₁ = 5 − 6 × 1 = −1

Putting n = 2, we get

a₂ = 5 − 6 × 2 = −7

Let d be the common difference of the AP.

∴ d = $a_2-a_1=-7-\left(-1\right)=-7+1=-6$

Sum of first n terms of the AP, S_n

$$\begin{gathered} =\frac{n}{2}\left[2\times \left(-1\right)+\left(n-1\right)\times \left(-6\right)\right]\left\{S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]\right\} \\ =\frac{n}{2}\left(-2-6n+6\right) \\ =n\left(2-3n\right) \\ =2n-3n^2 \end{gathered}$$

Putting n = 20, we get

$S_{20}=2\times 20-3\times {20}^2=40-1200=-1160$