Question

If given $5$ times the fifth term of an $A.P$. is equal to $8$ times its eight term, show that its ${13}^{th}$ term is zero.

Answer 1

Given $5$ times $5^{th}$ term $8$ times $8^{th}$ term

$5(a+(5-1\left)d\right)=8(a+(8-1\left)d\right)$ ......... $[t_n=a+(n-1\left)d\right]$

$5a+20d=8a+56d$

$\Rightarrow 3a+36d=0$

$\therefore a+12d=0$

Comparing this with $a+(n-1)d$,

we get ,$n-1=12$

$n={13}^{th}$ term.

Hence, ${13}^{th}$ term is zero.