CMO 2005 Problem 4

The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$ , and for $n\ge2$ ,$$2a_n-3a_{n-1}=\frac{3}{2^{n+1}}.$$Let $m$ be an integer with $m\ge2$ . Prove that: for $n\le m$ , we have$$\left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}.$$