Question

Choose the correct answer in each of the following questions:

The 5th term of an AP is 20 and the sum of its 7th and 11th terms is 64. The common difference of the AP is [CBSE 2015]

(a) 4 (b) 5 (c) 3 (d) 2

Answer 1

Let a be the first term and d be the common difference of the AP. Then,

$$\begin{gathered} a_5=20 \\ \Rightarrow a+\left(5-1\right)d=20\left[a_n=a+\left(n-1\right)d\right] \\ \Rightarrow a+4d=20.....\left(1\right) \end{gathered}$$

Now,

$a_7+a_{11}=64$ (Given)
$$\begin{gathered} \Rightarrow \left(a+6d\right)+\left(a+10d\right)=64 \\ \Rightarrow 2a+16d=64 \\ \Rightarrow a+8d=32.....\left(2\right) \end{gathered}$$

From (1) and (2), we get

$$\begin{gathered} 20-4d+8d=32 \\ \Rightarrow 4d=32-20=12 \\ \Rightarrow d=3 \end{gathered}$$

Thus, the common difference of the AP is 3.

Hence, the correct answer is option C.