Question

The sum of 3rd & 7th term of an AP is 6 and their product is 8 Find the sum of of first 16th term of the AP​

Answer 1

Given :-
→ a₃ + a₇ = 6.
→ a₃ × a₇ = 8 .
To find :-
→ S₁₅ .
Solution :-
We have,
→ a₃ + a₇ = 6.
⇒ a + 2d + a + 6d = 6 .
⇒ 2a + 8d = 6 .
⇒ 2( a + 4d ) = 6 .
⇒ a + 4d = 6/2 .
⇒ a + 4d = 3
∵a = 3 - 4d ............(1) .
And,
→ a₃ × a₇ = 8 .
⇒ ( a + 2d ) × ( a + 6d ) = 8 .
⇒ ( 3 - 4d + 2d ) ( 3 - 4d + 6d ) = 8 .
⇒ ( 3 - 2d )( 3 + 2d ) = 8 .
⇒ 3² - (2d)² = 8 .
⇒ 9 - 4d² = 8.
⇒ 4d² = 9 - 8 .
⇒ 4d² = 1 .
⇒ d² = 1/4 .
⇒ d = √(1/4) .
∴ d = 1/2 .
Putting the value of d in equation (1), we get ,
⇒ a = 3 - 4 × 1/2 .
⇒ a = 3 - 2 .
∴ a = 1 .
Thus, sum of 16th term is given by ,

Hence, sum of first 16 terms is 76.