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Standard XII

Mathematics

Property 4

Between 1 and...

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.

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Solution

Let A₁, A₂, … A_m be m numbers such that 1, A₁, A₂, … A_m, 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

A_m_–1 = a + (m – 1) d

According to the given condition,

Thus, the value of m is 14.

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Between $1$ and $31$ , $m$ numbers have been inserted in such a way that resulting sequence is an A.P. and the ratio of $7^{t h}$ and $(m - 1)^{t h}$ numbers (which are inserted) is $5 : 9$ . Find the value of $m$ .

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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.

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