Question

Find the ${31}^{\text{st}}$ term of an AP whose ${11}^{\text{th}}$ term is 38 and ${16}^{\text{th}}$ term is 73.

• A. 150
• B. 178
• C. 210
• D. 185

Answer 1

The correct option is B

178

Given that
$a_{11}=38$ and $a_{16}=73,$
where $a_{11}$ is the ${11}^{\text{th}}$ term and $a_{16}$ is the ${16}^{\text{th}}$ term of an AP.

The $n^{\text{th}}$ term of an AP with first term $a$ and common difference $d$ is given by
$a_n=a+(n-1)d.$
$\Rightarrow 38=a+10d....\left(i\right)$
$73=a+15d....\left(ii\right)$

Subtracting equation (i) from equation (ii), we get
$35=5d.$
$\Rightarrow d=7$

Substituting the value of $d$ in equation (i), we get
$a=38-10d=38-10\left(7\right)=-32.$

Now, $a_{31}=a+(31-1)d=-32+30\left(7\right)$
$=178$
Therefore, the ${31}^{\text{st}}$ term of the given AP is 178.