# Solving Т(n)=(1)/((1)/(T(n-1))+n^2) and T(1)=1 How do I go on from here? How can I find the solution? Thanks. :)

Solving $Т\left(n\right)=\frac{1}{\frac{1}{T\left(n-1\right)}+{n}^{2}}$ and $T\left(1\right)=1$
I did find $T\left(2\right)=\frac{1}{1+{n}^{2}}$, but I don't know how to proceed.
How do I go on from here? How can I find the solution? Thanks. :)
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Helena Howard
$\frac{1}{{t}_{n}}=\frac{1}{{t}_{n-1}}+{n}^{2}$
and the rest is smooth I think:
$\frac{1}{{t}_{2}}=\frac{1}{{t}_{1}}+{2}^{2}$
$\frac{1}{{t}_{3}}=\frac{1}{{t}_{2}}+{3}^{2}...$
$\frac{1}{{t}_{n}}=\frac{1}{{t}_{n-1}}+{n}^{2}.$
A summing of these equalities gives:
$\frac{1}{{t}_{n}}-\frac{1}{{t}_{1}}={2}^{2}+{3}^{2}+...+{n}^{2}$
or
$\frac{1}{{t}_{n}}={1}^{2}+{2}^{2}+{3}^{2}+...+{n}^{2}$
or
$\frac{1}{{t}_{n}}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}.$
${t}_{n}=\frac{6}{n\left(n+1\right)\left(2n+1\right)}$