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^(t h)` and `(m-1)^(t h)` numbers is 5 : 9. Find the value of m.

Answer

N/a

Explanation

`"Let "A_(1),A_(2),A_(3),…,A_(m)` be A.M.s between 1 and 31.
`therefore" "1, A_(1),A_(2),…,A_(m),` 31 are in A.P and number of terms in it are (m+2).
Hera a=1
`"and "T_(m+2)=31`
`rArr" "1+(m+1)d=31`
`rArr" "d=(30)/(m+1)" ...(1)"`
Given that,
`(A_(7))/(A_(m-1))=(5)/(9)rArr" "(T_(8))/(T_(m))=(5)/(9)`
`rArr" "(a+7d)/(a+(m-1)d)=(5)/(9)rArr" "(1+7d)/(1+(m-1)d)=(5)/(9)`
`rArr" "9+63d=5+(5m-5)d`
`rArr" "4=(5m-68)d`
`rArr" "4=(5m-68)cdot(30)/(m+1)`
`rArr" "4m+4=150m-2040`
`rArr" "146m=2044`
`rArr" "m=14`