**Homework Assignment 4***Course: COT3100 – Fall 2020*1) **Determine the following summation in terms of $ n $:** \[ \sum_{i=1}^{n^2} (2i + 5) \]2) **Let $ g(n) $ be a function defined on the positive integers as follows:** \[ g(1) = 3, \quad g(2) = 2, \quad g(3) = 4 \] For $ n > 3 $, \[ g(n) = g(n-1) + 2g(n-2) + 3g(n-3) \] What are the values of $ g(4) $, $ g(5) $, and $ g(6) $? *If you would like to, for fun, write a computer program which prints out the first 1000 values of $ g(n) \mod 10^9+7 $. Feel free to include the source code inside the document your homework solutions.*3) **Let $ n $ be a positive integer such that $ 7 \mid (8^n - 1) $. Prove that $ 7 \mid (8^{n+1} - 1) $. (Hint: Use a rule of powers to rewrite $ 8^{n+1} $.**4) **Prove by induction that, for all positive integers $ n $:** \[ \sum_{i=1}^{n} (3i^2 + 8i + 4) = \frac{2n^3 + 11n^2 + 17n}{2} \]5) **Recall that the Fibonacci numbers are defined as follows:** \[ F_0 = 0, \quad F_1 = 1 \] For $ n > 1 $, \[ F_n = F_{n-1} + F_{n-2} \] Using induction on $ n $, prove: \[ \sum_{i=0}^{n} F_i = F_{n+2} - 1 \]6) **Give a summary of the academic contributions of Dr

Answered: Determine the following summation in…

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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**Homework Assignment 4**

*Course: COT3100 – Fall 2020*

1) **Determine the following summation in terms of $ n $:**

\[
\sum_{i=1}^{n^2} (2i + 5)
\]

2) **Let $ g(n) $ be a function defined on the positive integers as follows:**

\[
g(1) = 3, \quad g(2) = 2, \quad g(3) = 4
\]

For $ n > 3 $,
\[
g(n) = g(n-1) + 2g(n-2) + 3g(n-3)
\]

What are the values of $ g(4) $, $ g(5) $, and $ g(6) $?

*If you would like to, for fun, write a computer program which prints out the first 1000 values of $ g(n) \mod 10^9+7 $. Feel free to include the source code inside the document your homework solutions.*

3) **Let $ n $ be a positive integer such that $ 7 \mid (8^n - 1) $. Prove that $ 7 \mid (8^{n+1} - 1) $. (Hint: Use a rule of powers to rewrite $ 8^{n+1} $.**

4) **Prove by induction that, for all positive integers $ n $:**

\[
\sum_{i=1}^{n} (3i^2 + 8i + 4) = \frac{2n^3 + 11n^2 + 17n}{2}
\]

5) **Recall that the Fibonacci numbers are defined as follows:**

\[
F_0 = 0, \quad F_1 = 1
\]

For $ n > 1 $,
\[
F_n = F_{n-1} + F_{n-2}
\]

Using induction on $ n $, prove:

\[
\sum_{i=0}^{n} F_i = F_{n+2} - 1
\]

6) **Give a summary of the academic contributions of Dr

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