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6. Use mathematical induction to prove that 2+5+8+11+.....+(3n-1)=n(3n+1)/2

6. Use mathematical induction to prove that 2+5+8+11+.....+(3n-1)=n(3n+1)/2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Discrete Mathematics: Please Help me with Question 6 (See attachment)

## Mathematical Problems and Solutions ### 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. Recursive Function Problem: For the following recursive functions, find \(f(1)\), \(f(2)\), \(f(3)\), and \(f(4)\) - Given: \(f(0) = 2\) \[f(k) = k - (f(k-1))^2\] - Calculations: - \(F(1) = 1 - (f(0))^2 = 1 - 4 = -3\) - \(F(2) = 2 - (f(1))^2 = 2 - (-3)^2 = -7\) - \(F(3) = 3 - (f(2))^2 = 3 - 49 = -46\) - \(F(4) = 4 - (f(3))^2 = 4 - (−46)^2 = 4 - 2116 = -2112\) ### 3. Prove Relation: 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. Solve the Recursive Relation Using Pattern Recognition: Given: - \(f(0) = 4\) \[f(k) = 5 + 1.1f(k-1) \quad \text{for } k > 0\] a) Evaluate \(f(10)\) ### 5. Divisibility Problem: Prove: \(n^3 + 5n\) is divisible by 6 for all integer \(n \geq 0\). ### 6. Mathematical Induction: Prove by induction: \[2 + 5 + 8 + 11 + \ldots + (
Transcribed Image Text:## Mathematical Problems and Solutions ### 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. Recursive Function Problem: For the following recursive functions, find \(f(1)\), \(f(2)\), \(f(3)\), and \(f(4)\) - Given: \(f(0) = 2\) \[f(k) = k - (f(k-1))^2\] - Calculations: - \(F(1) = 1 - (f(0))^2 = 1 - 4 = -3\) - \(F(2) = 2 - (f(1))^2 = 2 - (-3)^2 = -7\) - \(F(3) = 3 - (f(2))^2 = 3 - 49 = -46\) - \(F(4) = 4 - (f(3))^2 = 4 - (−46)^2 = 4 - 2116 = -2112\) ### 3. Prove Relation: 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. Solve the Recursive Relation Using Pattern Recognition: Given: - \(f(0) = 4\) \[f(k) = 5 + 1.1f(k-1) \quad \text{for } k > 0\] a) Evaluate \(f(10)\) ### 5. Divisibility Problem: Prove: \(n^3 + 5n\) is divisible by 6 for all integer \(n \geq 0\). ### 6. Mathematical Induction: Prove by induction: \[2 + 5 + 8 + 11 + \ldots + (
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