Question

If 7 times the 7^th term of an A.P. is equal to 11 times its 11^th term, then the value of its 18^th term is ________.

Answer 1

Given:
$7a_7=11a_{11}$

Now,
$$\begin{gathered} a_n=a+\left(n-1\right)d \\ \mathrm{Thus}, \\ 7a_7=11a_{11} \\ \Rightarrow 7\left[a+\left(7-1\right)d\right]=11\left[a+\left(11-1\right)d\right] \\ \Rightarrow 7\left[a+6d\right]=11\left[a+10d\right] \\ \Rightarrow 7a+42d=11a+110d \\ \Rightarrow 11a-7a=42d-110d \\ \Rightarrow 4a=-68d \\ \Rightarrow a=-17d...\left(1\right) \\ {a}_{18}=a+\left(18-1\right)d \\ =-17d+17d\left(\mathrm{From}\left(1\right)\right) \\ =0 \end{gathered}$$

Hence, the value of its 18^th term is 0.