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5. Prove: n + 5nis divisible by 6 for all integer n20.

5. Prove: n + 5nis divisible by 6 for all integer n20.

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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Discrete Mathematics: Please Help me with Question 5 (See attachment)

## Mathematical Exercises ### 1. Compute the following: a) \(\prod_{n=2}^{58} (-1)^n\) b) \(2 + 5 + 8 + 11 + \ldots + 302 + 305\) c) \(\sum_{i=0}^{100} \frac{i}{1000!}\) d) \(\frac{1000!}{998!}\) ### 2. For the following recursive functions, find \(f(1)\), \(f(2)\), \(f(3)\), and \(f(4)\): \(f(0) = 2\) \(f(k) = k - (f(k-1))^2\) - \(f(1) = 1 - (f(0))^2 = 1 - 4 = -3\) - \(f(2) = 2 - (f(1))^2 = 2 - (-3)^2 = 5\) - \(f(3) = 3 - (f(2))^2 = 3 - 25 = -22\) - \(f(4) = 4 - (f(3))^2 = 4 - (-22)^2 = 4 - 484 = -400\) ### 3. Prove that \(S_n = \frac{n^2 + n}{2}\) is a solution of the recursive relation \(S_1 = 1\) \(S_k = S_{k-1} + k \quad \text{for } k > 1\) ### 4. Using pattern recognition, solve the recursive relation a) \(f(0) = 4\) \(f(k) = 5 + 1.1 f(k-1) \quad \text{for } k > 0\) b) Evaluate \(f(10)\) ### 5. Prove: \(n^3 + 5n\) is divisible by 6 for all integer \(n \geq 0\). ### 6. Use mathematical induction to prove that \(2 + 5 + 8 + 11 + \ldots + (3n - 1) = n(
Transcribed Image Text:## Mathematical Exercises ### 1. Compute the following: a) \(\prod_{n=2}^{58} (-1)^n\) b) \(2 + 5 + 8 + 11 + \ldots + 302 + 305\) c) \(\sum_{i=0}^{100} \frac{i}{1000!}\) d) \(\frac{1000!}{998!}\) ### 2. For the following recursive functions, find \(f(1)\), \(f(2)\), \(f(3)\), and \(f(4)\): \(f(0) = 2\) \(f(k) = k - (f(k-1))^2\) - \(f(1) = 1 - (f(0))^2 = 1 - 4 = -3\) - \(f(2) = 2 - (f(1))^2 = 2 - (-3)^2 = 5\) - \(f(3) = 3 - (f(2))^2 = 3 - 25 = -22\) - \(f(4) = 4 - (f(3))^2 = 4 - (-22)^2 = 4 - 484 = -400\) ### 3. Prove that \(S_n = \frac{n^2 + n}{2}\) is a solution of the recursive relation \(S_1 = 1\) \(S_k = S_{k-1} + k \quad \text{for } k > 1\) ### 4. Using pattern recognition, solve the recursive relation a) \(f(0) = 4\) \(f(k) = 5 + 1.1 f(k-1) \quad \text{for } k > 0\) b) Evaluate \(f(10)\) ### 5. Prove: \(n^3 + 5n\) is divisible by 6 for all integer \(n \geq 0\). ### 6. Use mathematical induction to prove that \(2 + 5 + 8 + 11 + \ldots + (3n - 1) = n(
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