Question

Between $1$ and $31$ are inserted $m$ arithmetic means, so that the ratio of the $7^{\text {th }}$ and $(m-1)$ th means is $5: 9$. Then find the value of $m$.

Answer 1

$d=\frac{31-1}{m+1}=\frac{30}{m+1}$
$\frac{A_{7}}{A_{m-1}}=\frac{1+7 d}{31-2 d}=\frac{5}{9}$
$\Rightarrow d=2=\frac{30}{m+1} $
$\Rightarrow m=14$