Question

Find the relation between $x$ and $y$ in order that the $r^{th}$ mean between $x$ and $2y$ may be the same as the $r^{th}$ mean between $2x$ and $y$, $n$ means being inserted in each case.

Answer 1

When $n$ means are inserted between $a$ and $b$, common difference $d=\frac{b-a}{n+1}$ and $r^{\text{th}}$ mean $=a+rd$

Therefore $x+r\times \left(\frac{2y-x}{n+1}\right)=2x+r\times \left(\frac{y-2x}{n+1}\right)$

$⟹(n+1)x+2ry-rx=2(n+1)x+ry-2rx⟹y=\left(\frac{n-r+1}{r}\right)x$