Question

The sum of the $4^{\text{th}}$ and $8^{\text{th}}$ term of an A.P is $24$ and the sum of its $6^{\text{th}}$ and ${10}^{\text{th}}$ terms is $44$. Find the first term?

• A. $13$
• B. $12$
• C. $-13$
• D. $-14$

Answer 1

Answer: (C)

Case1:

Sum of the $4^{\text{th}}$ and $8^{\text{th}}$ terms of an A.P is $24$.

$t_4+t_8=a+3d+a+7d=2a+10d$

i.e., $2a+10d=24$

$\Rightarrow 2(a+5d)=24$

$\Rightarrow a+5d=12$ ---(1)

Case2:

Sum of the $6^{\text{th}}$ and ${10}^{\text{th}}$ terms of an A.P is $44$.

$t_6+t_{10}=a+5d+a+9d=44$

$\Rightarrow 2a+14d=44$

$\Rightarrow 2(a+7d)=44$

$\Rightarrow a+7d=22$ ---(2)

Lets do (1)-(2)

$a+5d-(a+7d)=12-22$

$\Rightarrow a+5d-a-7d=-10$

$\Rightarrow -2d=-10$

$\Rightarrow d=5$

Lets substitute $d=5$ in in $a+7d=22$

$\Rightarrow a+7\left(5\right)=22$

$\Rightarrow a+35=22$

$\Rightarrow a=22-35$

$\Rightarrow a=-13$
Hence, Option $D$ is correct.