Question

If 10 times the 10th term of an AP is equal to 15 times the 15th term, show that its 25th term is zero.

Answer 1

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

$10\times a_{10}=15\times a_{15}$ (Given)
$$\begin{gathered} \Rightarrow 10\left(a+9d\right)=15\left(a+14d\right)\left[a_n=a+\left(n-1\right)d\right] \\ \Rightarrow 2\left(a+9d\right)=3\left(a+14d\right) \\ \Rightarrow 2a+18d=3a+42d \\ \Rightarrow a=-24d \end{gathered}$$
$$\begin{gathered} \Rightarrow a+24d=0 \\ \Rightarrow a+\left(25-1\right)d=0 \\ \Rightarrow a_{25}=0 \end{gathered}$$

Hence, the 25th term of the AP is 0.