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.

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 7 th and (m – 1) th numbers is 5 : 9. Find the value of m.

Philoid · Accepted Answer

Let A₁, A₂, … A_m be m numbers such that 1, A₁, A₂, … A_m, 31 is an A.P.

Here, First term = a = 1,

Last term = b = 31,

Total no. of terms = 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

A_m-1 = a + (m – 1)d

Also given,

⇒

⇒ 9 (m + 211) = 5 (31m – 29)

⇒ 9m + 1899 = 155m - 145

⇒ 155m – 9m = 1899 + 145

⇒ 146m = 2044

∴ m = 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.