Let a 1 = 8 , a 2 , a 3 , … . a n be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170 , then the product of its middle two terms is _____ .

Question

TestHub · Accepted Answer

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

$a_{1} + a_{2} + a_{3} + a_{4} = 50$

$\Rightarrow 4 a + 6 d = 50$

$\Rightarrow 32 + 6 d = 50$

$\Rightarrow d = 3$

Also,

$a_{n - 3} + a_{n - 2} + a_{n - 1} + a_{n} = 170$

$\Rightarrow 4 a + (4 n - 10) \cdot d = 170$

$\Rightarrow 32 + (4 n - 10) \cdot 3 = 170$

$\Rightarrow n = 14$

So, middle terms are

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

$\Rightarrow a_{7} \cdot a_{8} = 754$

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