91Ó°ÊÓ

Prove the following statements with either induction, strong induction or proof by smallest counterexample.Prove that \(1+2+3+4+\cdots+n=\frac{n^{2}+n}{2}\) for every positive integer \(n\).

Use the method of direct proof to prove the following statements. Prove that \(1^{2}+2^{2}+3^{2}+4^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6}\) for every positive integer \(n .\)

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Prove that \(1^{3}+2^{3}+3^{3}+4^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}\) for every positive integer \(n\).

Prove the following statements with either induction, strong induction or proof by smallest counterexample. If \(n \in \mathbb{N},\) then \(1 \cdot 2+2 \cdot 3+3 \cdot 4+4 \cdot 5+\cdots+n(n+1)=\frac{n(n+1)(n+2)}{3}\).

Prove the following statements with either induction, strong induction or proof by smallest counterexample. If \(n \in \mathbb{N}\), then \(2^{1}+2^{2}+2^{3}+\cdots+2^{n}=2^{n+1}-2\).

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Prove that \(\sum_{i=1}^{n}(8 i-5)=4 n^{2}-n\) for every positive integer \(n\).

Prove the following statements with either induction, strong induction or proof by smallest counterexample. If \(n \in \mathbb{N},\) then \(1 \cdot 3+2 \cdot 4+3 \cdot 5+4 \cdot 6+\cdots+n(n+2)=\frac{n(n+1)(2 n+7)}{6}\).

Prove the following statements with either induction, strong induction or proof by smallest counterexample. If \(n \in \mathbb{N},\) then \(\frac{1}{2 !}+\frac{2}{3 !}+\frac{3}{4 !}+\cdots+\frac{n}{(n+1) !}=1-\frac{1}{(n+1) !}\)

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Prove that \(24 \mid\left(5^{2 n}-1\right)\) for every integer \(n \geq 0\).

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Prove that \(3 \mid\left(5^{2 n}-1\right)\) for every integer \(n \geq 0\).