8. Solve the recursion: A = 1; A2 = -1 Ak = 5Ak-1-6Ar-2 %3D
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Discrete Mathematics: Please Help me with Question 8 (See attachment)
![### Educational Mathematics Problems
#### 1. Compute the following:
a) \(\prod_{n=1}^{58} (-1)^n\)
b) \(2+5+8+11+\ldots+302+305\)
c) \(\sum_{i=0}^{100} \frac{i}{1000!}\)
d) \(\frac{1000!}{998!}\)
#### 2. Recursive Functions
For the following recursive function, find \(f(1)\), \(f(2)\), \(f(3)\), and \(f(4)\):
\[f(0) = 2\]
\[f(k) = k - (f(k-1))^2\]
- \(F(1) = 1 - (f(0))^2 = 1 - 4 = -3\)
- \(F(2) = 2 - (f(1))^2 = 2 - (-3)^2 = 5\)
- \(F(3) = 3 - (f(2))^2 = 3 - 25 = -22\)
- \(F(4) = 4 - (f(4))^2 = 4 - (-22)^2 = 4 - 484 = -400\)
#### 3. Prove the Sequence Formula:
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. Recursive Relation Recognition:
Using pattern recognition, solve the recursive relation:
a) \(f(0) = 4\)
b) \(f(k) = 5 + 1.1f(k-1) \quad \text{for } k > 0\)
c) Evaluate \(f(10)\)
#### 5. Prove Divisibility:
Prove: \(n^3 + 5n\) is divisible by 6 for all integers \(n \geq 0\).
#### 6. Mathematical Induction:
Use mathematical induction to prove that:
\[2 + 5 + 8 + 11 + \ldots + (3n - 1) = n(3n +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6805e19-16c9-4240-8da2-d1d02f6f503a%2F06bb493f-7271-41eb-afc3-e2d584741afe%2F2372gob_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Mathematics Problems
#### 1. Compute the following:
a) \(\prod_{n=1}^{58} (-1)^n\)
b) \(2+5+8+11+\ldots+302+305\)
c) \(\sum_{i=0}^{100} \frac{i}{1000!}\)
d) \(\frac{1000!}{998!}\)
#### 2. Recursive Functions
For the following recursive function, find \(f(1)\), \(f(2)\), \(f(3)\), and \(f(4)\):
\[f(0) = 2\]
\[f(k) = k - (f(k-1))^2\]
- \(F(1) = 1 - (f(0))^2 = 1 - 4 = -3\)
- \(F(2) = 2 - (f(1))^2 = 2 - (-3)^2 = 5\)
- \(F(3) = 3 - (f(2))^2 = 3 - 25 = -22\)
- \(F(4) = 4 - (f(4))^2 = 4 - (-22)^2 = 4 - 484 = -400\)
#### 3. Prove the Sequence Formula:
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. Recursive Relation Recognition:
Using pattern recognition, solve the recursive relation:
a) \(f(0) = 4\)
b) \(f(k) = 5 + 1.1f(k-1) \quad \text{for } k > 0\)
c) Evaluate \(f(10)\)
#### 5. Prove Divisibility:
Prove: \(n^3 + 5n\) is divisible by 6 for all integers \(n \geq 0\).
#### 6. Mathematical Induction:
Use mathematical induction to prove that:
\[2 + 5 + 8 + 11 + \ldots + (3n - 1) = n(3n +
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