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Problem 18

Prove by mathematical induction $1^{2}+2^{2}+3^{2}+\ldots+n^{2}=(1 / 6) n(n+1)(2 n+1)$.

Expert verified

By mathematical induction, we proved that the formula for the sum of the first n squares is true for all positive integers n:
\(1^2+2^2+3^2+\ldots+n^2 = \frac{1}{6} n(n+1)(2n+1)\).
The proof consists of the following steps:
1. Verify the base case (n=1).
2. State the induction hypothesis for n=k.
3. Prove the induction step for n=k+1.
4. Simplify the RHS of the equation.
5. Relate the simplified RHS to the desired form.
6. Conclude that the formula holds true for all positive integers n.

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