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 \(−3\).