If the sum of the series 20 + 19 3 5 + 19 1 5 + 18 4 5 + . . . . . . . . . . up to n th term is 488 and the n th term is negative, then :

Step-by-step solution

The given series is an arithmetic progression whose first term is 20 and common difference is 19 3 5 - 20 = - 2 5 . According to question, 488 = n 2 2 20 + n - 1 - 2 5 ⇒ 488 = n 2 40 - 2 n 5 + 2 5 ⇒ n 2 - 101 n + 2440 = 0 ⇒ n = 61 or 40 For n = 40 ⇒ T n = T 40 = 20 + 40 - 1 - 2 5 = 22 5 > 0 For n = 61 ⇒ T n = T 61 = 20 + 61 - 1 - 2 5 = - 4 < 0 Thus, T n = T 61 = - 4