Suppose a 1 , a 2 , … , a n , … be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is 5 : 17 and 110 < a 15 < 120 , then the sum of the first ten terms of the progression is equal to

Step-by-step solution

Given the ratio of the sum of the first five terms to the sum of the first nine terms is 5 : 17 , So, S 5 S 9 = 5 17 ⇒ 5 2 2 a + 4 d 9 2 2 a + 8 d = 5 17 ⇒ 17 2 a + 4 d = 9 2 a + 8 d ⇒ 34 a + 68 d = 18 a + 72 d ⇒ d = 4 a Now a 15 = a + 14 d = 57 a Also given 110 < a 15 < 120 ⇒ 110 < 57 a < 120 ⇒ a = 2     ∴   d = 8 So, S 10 = 10 2 2 × 2 + 9 × 8 = 380