Let a _ 1 = 8, a _ 2 , a _ 3 , . a _ n be an A.P. If the sum of its...

Let the common difference of an A.P. be $d$, then

${a}_{1} + {a}_{2} + {a}_{3} + {a}_{4} = 50$

$⇒ 4a + 6d = 50$

$⇒ 32 + 6d = 50$

$⇒ d = 3$

Also,

${a}_{n - 3} + {a}_{n - 2} + {a}_{n - 1} + {a}_{n} = 170$

$⇒ 4a + \left(4n - 10\right) \cdot d = 170$

$⇒ 32 + \left(4n - 10\right) \cdot 3 = 170$

$⇒ n = 14$

So, middle terms are

${a}_{7} = a + 6d = 26, {a}_{8} = a + 7d = 29$

$⇒ {a}_{7} \cdot {a}_{8} = 754$