Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A - Maths - Sequences and Series

Question

Between 1 and 31, m numbers have been inserted in such a
way that the resulting sequence is an A P and the ratio of
7th and (m – 1)th numbers is 5:9
Find the value of m

Gopal Mohanty · Accepted Answer

Let A1, A2, … Am be m numbers such that 1, A1, A2, … Am, 31 is an A P Here, a = 1, b = 31, n = m + 2 ∴ 31 = 1 + (m + 2 – 1) (d) ⇒ 30 = (m + 1) d

A1 = a + d A2 = a + 2d A3 = a + 3d … ∴ A7 = a + 7d Am–1 = a + (m – 1) d According to the given condition,

Thus, the value of m is 14

Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.

Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7^th and (m – 1)^th numbers is 5:9. Find the value of m.