1. a) Prove by mathematical induction that $$ 1^{2}+2^{2}+\ldots+n^{2}=n(n+1)(2 n+1) / 6 $$ for all natural numbers $$ n $$. b) As an alternative method, we know the sum can be expressed as $$ A n^{3}+B n^{2}+C n $$ for appropriate constants $$ A, B $$ and $$ C $$. Write down three equations that can be solved to determine $$ A, B $$ and $$ C $$. c) Using a), what are the values of $$ A, B $$, and $$ C $$ ?

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a) The formula $$ 1^2 + 2^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6} $$ is proven for all natural numbers $$ n $$ using mathematical induction. b) The equations to determine $$ A, B, $$ and $$ C $$ are: 1. $$ A + B + C = 1 $$ 2. $$ 8A + 4B + 2C = 5 $$ 3. $$ 27A + 9B + 3C = 14 $$ c) The values of $$ A, B, $$ and $$ C $$ are: $$ A = \frac{1}{3}, \quad B = \frac{1}{2}, \quad C = \frac{1}{6} $$

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