Question

The $4^{th}$ term of an AP is zero.Prove that the ${25}^{th}$ term is thrice its $11th$ term.

Answer 1

Let, $a$ and $d$ be the first term and common difference of the $A.P$ respectively.

According to the question, $a_4=0\Rightarrow a+3d=0\Rightarrow a=-3d$

Now, the ${25}^{th}$ term, $a_{25}=a+24d$

the ${11}^{th}$ term, $a_{11}=a+10d$

and, $\frac{a_{25}}{a_{11}}=\frac{a+24d}{a+10d}\Rightarrow \frac{a_{25}}{a_{11}}=\frac{-3d+24d}{-3d+10d}\Rightarrow \frac{a_{25}}{a_{11}}=\frac{21d}{7d}\Rightarrow \frac{a_{25}}{a_{11}}=3\Rightarrow a_{25}=3\times a_{11}$

Hence, the ${25}^{th}$ term is thrice of the ${11}^{th}$ of the $A.P.$