Discrete Mathematics: Please Help me with Question 9 (See attachments, they are split between the pages: parts a,b,c) Certainly, here is the transcription of the provided text, suitable for an educational website:---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) $f(0) = 2$ $f(k) = k - (f(k-1))^2$ \[ \begin{align*} f(1) &= 1 - (f(0))^2 = 1 - 4 = -3 \\ f(2) &= 2 - (f(1))^2 = 2 - (-3)^2 = 2 - 9 = -7 \\ f(3) &= 3 - (f(2))^2 = 3 - 49 = -46 \\ f(4) &= 4 - (f(3))^2 = 4 - (-46)^2 = 4 - 2116 = -2112 \end{align*} \]3. Prove that $S_n = \frac{n^2 + n}{2}$ is a solution of the recursive relation \[ \begin{align*} S_1 &= 1 \\ S_k &= S_{k-1} + k \quad \text{for} \, k > 1 \end{align*} \]4. 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: $n^3 + 5n$ is divisible by 6 for all integer $n \geq 0$.6. Use mathematical induction to prove that $2+5+8 Certainly, here's a transcription suitable for an educational website:---**Mathematics Problem Set****c) Find the Inverse of \( f(x) $**---**10. Set Theory Problem**Define the sets:- $ S = \{x \mid x \mod 4 = 3\} $- $ T = \{y \mid y \mod 2 = 1\} $Prove that $ S $ is a subset of $ T $.--- ### ExplanationIn this problem, we are asked to prove that every element of set $ S $ is also an element of set $ T $.1. **Set $ S $:** Consists of numbers $ x $ such that when divided by 4, the remainder is 3.2. **Set $ T $:** Consists of numbers $ y $ such that when divided by 2, the remainder is 1 (odd numbers).**Proof Strategy:**- When a number $ x $ has a remainder of 3 when divided by 4, it can be expressed in the form $ x = 4k + 3 $ where $ k $ is an integer.- This number $ x = 4k + 3 $ will have a remainder of 1 when divided by 2, indicating that it is an odd number and thus an element of $ T $. Thus, $ S \subseteq T $.

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Answered: Discrete Mathematics: Please Help me…

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Discrete Mathematics: Please Help me with Question 9 (See attachments, they are split between the pages: parts a,b,c)

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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Concept explainers

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

Discrete Mathematics: Please Help me with Question 9 (See attachments, they are split between the pages: parts a,b,c)

Certainly, here is the transcription of the provided text, suitable for an educational website:

---

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)

$f(0) = 2$

$f(k) = k - (f(k-1))^2$

\[
\begin{align*}
f(1) &= 1 - (f(0))^2 = 1 - 4 = -3 \\
f(2) &= 2 - (f(1))^2 = 2 - (-3)^2 = 2 - 9 = -7 \\
f(3) &= 3 - (f(2))^2 = 3 - 49 = -46 \\
f(4) &= 4 - (f(3))^2 = 4 - (-46)^2 = 4 - 2116 = -2112
\end{align*}
\]

3. Prove that $S_n = \frac{n^2 + n}{2}$ is a solution of the recursive relation

\[
\begin{align*}
S_1 &= 1 \\
S_k &= S_{k-1} + k \quad \text{for} \, k > 1
\end{align*}
\]

4. 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: $n^3 + 5n$ is divisible by 6 for all integer $n \geq 0$.

6. Use mathematical induction to prove that

\(2+5+8

Certainly, here's a transcription suitable for an educational website:

---

**Mathematics Problem Set**

**c) Find the Inverse of $ f(x) $**

---

**10. Set Theory Problem**

Define the sets:
- $ S = \{x \mid x \mod 4 = 3\} $
- $ T = \{y \mid y \mod 2 = 1\} $

Prove that $ S $ is a subset of $ T $.

---

### Explanation

In this problem, we are asked to prove that every element of set $ S $ is also an element of set $ T $.

1. **Set $ S $:** Consists of numbers $ x $ such that when divided by 4, the remainder is 3.

2. **Set $ T $:** Consists of numbers $ y $ such that when divided by 2, the remainder is 1 (odd numbers).

**Proof Strategy:**

- When a number $ x $ has a remainder of 3 when divided by 4, it can be expressed in the form $ x = 4k + 3 $ where $ k $ is an integer.
- This number $ x = 4k + 3 $ will have a remainder of 1 when divided by 2, indicating that it is an odd number and thus an element of $ T $.

Thus, $ S \subseteq T $.

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