Question

In an AP the first term is 22, nth term is −11 and sum to first nth terms is 66. Find n and d, the common difference.

Answer 1

Here, a = 22, T_n = -11 and S_n = 66
Let d be the common difference of the given AP.
Then T_n = -11
⇒ a + (n - 1)d = 22 + (n - 1)d = -11
⇒ (n - 1)d = -33 ...(i)

The sum of n terms of an AP is given by
$$\begin{gathered} S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]=66 \\ \Rightarrow \frac{n}{2}\left[2\times 22+(-33)\right]=\left(\frac{n}{2}\right)\times 11=66 \\ \Rightarrow n=12 \end{gathered}$$ [Substituting the value of (n - 1)d from (i)]
Putting the value of n in (i), we get:
11d = -33
⇒ d = -3
Thus, n = 12 and d = -3