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Prove the equality 1^(2)+2^(2)+3^(2) ...

Prove the equality
`1^(2)+2^(2)+3^(2) . . .+n^(2)=.^(n+1)C_(2)+2(.^(n)C_(2)+.^(n-1)C_(2) . . .+.^(2)C_(2))`.

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To prove the equality \[ 1^2 + 2^2 + 3^2 + \ldots + n^2 = \binom{n+1}{2} + 2 \left( \binom{n}{2} + \binom{n-1}{2} + \ldots + \binom{2}{2} \right) \] we will start by calculating both sides of the equation. ### Step 1: Calculate the Left-Hand Side (LHS) The left-hand side is the sum of the squares of the first \( n \) natural numbers. The formula for this sum is: \[ 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6} \]
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