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2n 10. Prove that 31 (5 - 1) for every integer n 2 0.

2n 10. Prove that 31 (5 - 1) for every integer n 2 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 10 please 

### Mathematical Induction and Proof Problems 1. **Prove that \(1^2 + 2^2 + 3^2 + 4^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\)** 2. **Prove that \(1^3 + 2^3 + 3^3 + 4^3 + \cdots + n^3 = \frac{n^2(n+1)^2}{4}\)** 3. **If \(n \in \mathbb{N}\), then \(1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot \cdots \cdot n(n+1) = n(n+1)\)** 4. **If \(n \in \mathbb{N}\), then \(2^1 + 2^2 + 2^3 + \cdots + 2^n = 2^{n+1} - 2\)** 5. **Prove that \(\sum_{i=1}^{n} (8i - 5) = 4n^2 - n\) for every positive integer \(n\)** 6. **If \(n \in \mathbb{N}\), then \(1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + 4 \cdot 6 + \cdots + n(n+2) = n(n + 2)\)** 7. **If \(n \in \mathbb{N}\), then \(\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \cdots + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}\)** 8. **Prove that \(24 \mid (5^{2n} - 1)\) for every integer \(n \geq 0\)** 9. **Prove that \(3 \mid (5^{2n} - 1)\) for every integer \(n \geq 0\)** 10. **Prove that \(3 \mid (n^3 + 5n + 6)\) for every
Transcribed Image Text:### Mathematical Induction and Proof Problems 1. **Prove that \(1^2 + 2^2 + 3^2 + 4^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\)** 2. **Prove that \(1^3 + 2^3 + 3^3 + 4^3 + \cdots + n^3 = \frac{n^2(n+1)^2}{4}\)** 3. **If \(n \in \mathbb{N}\), then \(1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot \cdots \cdot n(n+1) = n(n+1)\)** 4. **If \(n \in \mathbb{N}\), then \(2^1 + 2^2 + 2^3 + \cdots + 2^n = 2^{n+1} - 2\)** 5. **Prove that \(\sum_{i=1}^{n} (8i - 5) = 4n^2 - n\) for every positive integer \(n\)** 6. **If \(n \in \mathbb{N}\), then \(1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + 4 \cdot 6 + \cdots + n(n+2) = n(n + 2)\)** 7. **If \(n \in \mathbb{N}\), then \(\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \cdots + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}\)** 8. **Prove that \(24 \mid (5^{2n} - 1)\) for every integer \(n \geq 0\)** 9. **Prove that \(3 \mid (5^{2n} - 1)\) for every integer \(n \geq 0\)** 10. **Prove that \(3 \mid (n^3 + 5n + 6)\) for every
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