Question

Chapter 10: Prove the following statements with either induction, strong induction or proof by smallest counterexample. 2) Prove that $1^2 + 2^2 + 3^2 + 4^2 + dots + n^2 = frac{[n(n+1)(2n+1)]}{6}$ for every positive integer $n$. 10) Prove that $3 | (5^{2n} - 1)$ for every integer $n ge 0$. Proof 16) Prove that $2^n + 1 le 3^n$ for every positive integer $n$. 34) Prove that $3^1 + 3^2 + 3^3 + 3^4 + dots + 3^n = frac{(3^{n+1} - 3)}{2}$ for every $n in mathbb{N}$.

          Chapter 10:
Prove the following statements with either induction, strong induction or proof by smallest
counterexample.
2) Prove that $1^2 + 2^2 + 3^2 + 4^2 + dots + n^2 = frac{[n(n+1)(2n+1)]}{6}$ for every positive integer $n$.
10) Prove that $3 | (5^{2n} - 1)$ for every integer $n ge 0$.
Proof
16) Prove that $2^n + 1 le 3^n$ for every positive integer $n$.
34) Prove that $3^1 + 3^2 + 3^3 + 3^4 + dots + 3^n = frac{(3^{n+1} - 3)}{2}$ for every $n in mathbb{N}$.

$Chapter 10: Prove the following statements with either induction, strong induction or proof by smallest counterexample. 2) Prove that 1^2 + 2^2 + 3^2 + 4^2 + dots + n^2 = frac[n(n+1)(2n+1)]6 for every positive integer n. 10) Prove that 3 | (5^2n - 1) for every integer n ge 0. Proof 16) Prove that 2^n + 1 le 3^n for every positive integer n. 34) Prove that 3^1 + 3^2 + 3^3 + 3^4 + dots + 3^n = frac(3^n+1 - 3)2 for every n in mathbbN.$

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