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.

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.

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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  $−3$ .

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.

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