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Using pattern recognition, solve the recursive relation f(0) = 4 а) f(k) = 5+1.1f(k-1) for k>0 b) Evaluate f(10)

Using pattern recognition, solve the recursive relation f(0) = 4 а) f(k) = 5+1.1f(k-1) for k>0 b) Evaluate f(10)

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 4 (See attachment)

**Mathematics Problem Set** 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):** \( f(0) = 2 \) \( f(k) = k - (f(k-1))^2 \) - \( f(1) = 1 - (f(0))^2 = 1 - 4 = -3 \) - \( f(2) = 2 - (f(1))^2 = 2 - (-3)^2 = 2 - 9 = -7 \) - \( f(3) = 3 - (f(2))^2 = 3 - 49 = -46 \) - \( f(4) = 4 - (f(3))^2 = 4 - (-46)^2 = 4 - 2116 = -2112 \) 3. **Prove that \( S_n = \frac{n^2+n}{2} \) is a solution of the recursive relation:** - \( S_1 = 1 \) - \( S_k = S_{k-1} + k \) for \( k > 1 \) 4. **Using pattern recognition, solve the recursive relation:** - a) \( f(0) = 4 \) - \( f(k) = 5 + 1.1 f(k-1) \) 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:** \( 2 + 5 + 8 + 11 + \ldots + (3n-1) = n(3n+1)/2 \)
Transcribed Image Text:**Mathematics Problem Set** 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):** \( f(0) = 2 \) \( f(k) = k - (f(k-1))^2 \) - \( f(1) = 1 - (f(0))^2 = 1 - 4 = -3 \) - \( f(2) = 2 - (f(1))^2 = 2 - (-3)^2 = 2 - 9 = -7 \) - \( f(3) = 3 - (f(2))^2 = 3 - 49 = -46 \) - \( f(4) = 4 - (f(3))^2 = 4 - (-46)^2 = 4 - 2116 = -2112 \) 3. **Prove that \( S_n = \frac{n^2+n}{2} \) is a solution of the recursive relation:** - \( S_1 = 1 \) - \( S_k = S_{k-1} + k \) for \( k > 1 \) 4. **Using pattern recognition, solve the recursive relation:** - a) \( f(0) = 4 \) - \( f(k) = 5 + 1.1 f(k-1) \) 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:** \( 2 + 5 + 8 + 11 + \ldots + (3n-1) = n(3n+1)/2 \)
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