Triangular numbers

Algebra Level 5

Let \(T_1, T_2, ..., T_i\) be the first \(i\) triangular numbers. Evaluate the last five digits of:

\[ \sum_{i=1} ^{101} T_i T_{102-i} = T_1 T_{101} + T_2 T_{100}+...+T_{101} T_1 \]

Note: The \(n\)th triangular number is the sum of the first \(n\) natural numbers \(T_n = \dfrac{n(n+1)}{2}\). For instance \(T_5 = 1+2+3+4+5 = \dfrac{5 \cdot 6}{2} = 15\).

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