Between 1 and  31  are  inserted  M arithmetic means so that the ratio of the 7th  and  (M - 1) th  means is 5:9,  then the  value  of m is

let A1, A2,.,. , Am be ,m arithmetic means between 1 and 31. ⇒Then A1,A2,…,Am31 is an AP with common difference.d=31−1m+1=30m+1 ∵d=b−an+1∴  A7=1+7d=1+7×30m+1=m+211m+1and  Am−1=1+(m−1)d=1+30m+1(m−1)=31m−29m+1Now,  A7Am−1=59m+21131m−29=59⇒ 9m+1899=155m−145⇒ 146m=2044⇒ m=14 (given)