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\begin{align}
T_n + T_{n-1} &= \frac{n(n+1)}{2} + \frac{(n-1)n}{2}\\
&= \left (\frac{n^2}{2} + \frac{n}{2}\right) + \left(\frac{n^2}{2} - \frac{n}{2} \right )\\
&= n^2
\end{align}
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