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, $a_n = a + (n − 1) d$
$a_4 = a + (4 − 1) d$
$a_4 = a + 3d$
Similarly,
$a_8 = a + 7d$
$a_6 = a + 5d$
$a_{10} = a + 9d$
Given that, $a_4 + a_8 = 24$
$a + 3d + a + 7d = 24$
$2a + 10d = 24$
$a + 5d = 12$ $…….(1)$
$a_6 + a_{10} = 44$
$a + 5d + a + 9d = 44$
$2a + 14d = 44$
$a + 7d = 22$ $…….(2)$
On subtracting equation (1) from (2), we obtain
$2d = 22 − 12$
$2d = 10$
$d = 5$
From equation (1), we obtain
$a + 5d = 12$
$a + 5 (5) = 12$
$a + 25 = 12$
$a = −13$
$a_2 = a + d = − 13 + 5 = −8$
$a_3 = a_2 + d = − 8 + 5 = −3$

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