a_n=\frac{a_{n-1}+a_{n+1}}{2}
\frac{a_{n-1}+a_{n+1}}{2}=\frac{\frac{2(n-1)^2+8(n-1)+6}{n-1+1}+\frac{2(n+1)^2+8(n+1)+6}{n+1+1}}{2}=\frac{(n-1)^2+4(n-1)+3}{n}+\frac{(n+1)^2+4(n+1)+3}{n+2}=\frac{n^2-2n+1+4n-4+3}{n}+\frac{n^2+2n+1+4n+4+3}{n+2}=\frac{n^2+2n}{n}+\frac{n^2+6n+8}{n+2}=\frac{n(n+2)}{n}+\frac{n^2+6n+8}{n+2}=n+2+\frac{n^2+6n+8}{n+2}=\frac{(n+2)(n+2)+n^2+6n+8}{n+2}=\frac{n^2+4n+4+n^2+6n+8}{n+2}=
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