Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P

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.

Expert Solution

Let’s consider a1, a2, … am be m numbers such that 1, a1, a2, … am, 31 is an A.P.

and here,

a = 1, b = 31, n = m + 2

So, 31 = 1 + (m + 2 – 1) (d)

30 = (m + 1) d

d = 30/ (m + 1) ……. (1)

Now,

a1 = a + d

a2 = a + 2d

a3 = a + 3d …

Hence, a7 = a + 7d

am–1 = a + (m – 1) d

According to the question, we have

On substituting the known values we get m= 14

The value of m will be 14.

Nikhil Thakur answered 2 years ago

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