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*.

Let A_{1}, A_{2},
… A_{m} be *m* numbers such that 1, A_{1},
A_{2}, … 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*

A_{1} = *a*
+ *d*

A_{2} = *a*
+ 2*d*

A_{3} = *a*
+ 3*d* …

∴ A_{7}
= *a* + 7*d*

A_{m}_{–1}
= *a* + (*m* – 1)* d*

According to the given condition,

Thus, the value of* m*
is 14.

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