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.

Explanation

Let `A_(1),A_(2),…A_(m)` be m artimatic means between 1 and 31.
Then 1,`A_(1),A_(2),…A_(m)`,31 is an A.P..
Common difference,`d=(31-1)/(m+1)=30/(m+1)`
Now, `A_(7)=1+7d=1+(7xx30)/(m+1)=(m+211)/(m+1)`
and, `A_(m-1)=1+(m-1)d=1+30/(m+1)(m-1)=(31m-29)/(m+1)`
It is given that `(A_(7))/(A_(m-1))=5/9`
`rArr(m+211)/(31m-29)=5/9` ltbr gt`rArrm=14`