Question

The sum of first seven terms of an A.P. is 182. If its 4^th and the 17^th terms are in the ratio 1 : 5, find the A.P.

Answer 1

Let a be the first term and d be the common difference.

We know that, sum of first n terms = S_n = $\frac{n}{2}$[2a + (n − 1)d]

According to the question,

$$\begin{gathered} S_7=182 \\ \Rightarrow \frac{7}{2}\left[2a+\left(7-1\right)d\right]=182 \\ \Rightarrow \frac{1}{2}\left(2a+6d\right)=26 \\ \Rightarrow a+3d=26 \\ \Rightarrow a=26-3d....\left(1\right) \end{gathered}$$

Also,
$$\begin{gathered} \frac{a_4}{a_{17}}=\frac{1}{5} \\ \Rightarrow \frac{a+(4-1)d}{a+(17-1)d}=\frac{1}{5} \\ \Rightarrow \frac{a+3d}{a+16d}=\frac{1}{5} \\ \Rightarrow 5(a+3d)=a+16d \\ \Rightarrow 5a+15d=a+16d \\ \Rightarrow 5a-a=16d-15d \\ \Rightarrow 4a=d....\left(2\right) \end{gathered}$$

On substituting (2) in (1), we get
$$\begin{gathered} a=26-3\left(4a\right) \\ \Rightarrow a=26-12a \\ \Rightarrow 12a+a=26 \\ \Rightarrow 13a=26 \\ \Rightarrow a=2 \\ \Rightarrow d=4\times 2\left[\mathrm{From}\left(2\right)\right] \\ \Rightarrow d=8 \end{gathered}$$

Thus, the A.P. is 2, 10, 18, 26, ..... .