Question

The sum of the $4^{th}$ and $8^{th}$ terms of an AP is $24$ and the sum of the $6^{th}$ and ${10}^{th}$ terms is $44$. Find the first three terms of the AP.

Answer 1

We know that,

$a_n=a+(n-1)d$

$a_4=a+3d$ ---------------------- (1)

Similarly,

$a_8=a+7d$ -----------------------(2)

$a_6=a+5d$ -----------------------(3)

$a_{10}=a+9d$ ---------------------(4)

Given that,

$a_4+a_8=24$

$⟹a+3d+a+7d=24$ [From (1) amd (2)]
$⟹2a+10d=24$

$⟹a+5d=12$ ....(5)

Also, $a_6+a_{10}=44$

$⟹a+5d+a+9d=44$ [From (3) and (4)]

$⟹2a+14d=44$

$⟹a+7d=22$....(6)

On subtracting equation (5) from (6), we get

$2d=22-12$

$2d=10$

$d=5$

From equation (5), we get

$a+5d=12$

$a+5\left(5\right)=12$

$a+25=12$

$a=12-25=-13$

Lets evaluate second and third terms
$a_2=a+d$

$=-13+5=-8$

$a_3=a_2+d=-8+5=-3$

Therefore, the first three terms of this A.P. are $-13,-8$ and $-3$.