Question

The $8^{th}$ and ${24}^{th}$ terms of an A.P. are $45$ and $157$ respectively, find the ${33}^{rd}$ term.

Answer 1

Since,

$a_8=45$

$\Rightarrow a+7d=45⟶\left(i\right)\{\because a_n=a+(n-1)d\}$

$a_{24}=157\left\{Given\right\}$

$\Rightarrow a+23d=157⟶\left(ii\right)$

Subtracting ${eq}^n\left(i\right)$ from $\left(ii\right)$, we have

$a+23d-a-7d=157-45$

$16d=112$

$\Rightarrow d=7$

Substituting the value of $d$ in ${eq}^n\left(i\right)$, we have

$a+(7\times 7)=45$

$\Rightarrow a=45-49=-4$

$\therefore a_{33}=a+32d$

$\Rightarrow =-4+(32\times 7)$

$\Rightarrow =-4+224$

$\Rightarrow a_{33}=220$

Hence, the ${33}^{rd}$ term is $220$.