In a series of 100 numbers, Sn = 2n2 + 3, where Sn is the average of all the numbers from the nth term of the series. What is the value of the 16th term in the series?

In a series of 100 numbers, Sn = 2n2 + 3, where Sn is the average of all the numbers from the nth term of the series. What is the value of the 16th term in the series? Correct Answer -5029

Average of all the numbers from n^th term = Sum of all numbers from n^th term/ (100 – n + 1)

⇒ 2n² + 3 = Sum of all numbers from n^th term/ (100 – n + 1)

⇒ Sum of all numbers from n^th term = (2n² + 3) × (100 – n + 1)      ---- 1

Sum of all numbers from (n + 1)^th term = {2(n + 1)² + 3} × {100 – (n + 1) + 1}      ---- 2

⇒ n^th term = Sum of all numbers from n^th term – Sum of all the numbers from (n + 1)^th term.

For n = 16

Sum of all number from 16^th term from equation 1 = 43775

Sum of all numbers from 17^th term from equation 2 = 48804

∴ 16^th term = 43775 – 48804 = -5029