Question

In A.P., if $5$ times $5^{th}$ term $=7$ times $7^{th}$ term. Find ${12}^{th}$ term

Answer 1

According to the question,

$5a_5=7a_7$

$5\left[a+\left(5-1\right)d\right]=7\left[a+\left(7-1\right)d\right]$

$5\left(a+4d\right)=7\left(a+6d\right)$

$5a+20d=7a+42d$

$2a+22d=0$

$a+11d=0$

$a+\left(12-1\right)d=0$

$a_{12}=0$

Therefore, the ${12}^{th}$ term is 0.