CSCI 301 L16 Worksheet

Lecture 16/17 - Exercises

Part A - Proof by Induction

1. (Ch 10, Ex #1) Prove that \(1 + 2 + 3 + 4 + \cdots + n = \frac{n^2 + n}{2}\) for every positive integer \(n\).

2. Prove the following proposition: \[ \forall n \in \mathbb{N}, \left( \sum_{i=0}^n 2^i \right) = 2^{n+1} - 1 \]

3. (Ch 10, Ex #11) Prove that \(3 \mid (n^2 + 5n + 6)\) for every integer \(n \ge 0\).

Part B - More Proof by Induction

1. Prove that any integer numbers of cents \(\ge 8\) can be made using a combination of \(5\textcent\) and \(3\textcent\) coins.
2. (Ch10, Ex #25) Let \(F_i\) be the \(i\)th element of the Fibonacci sequence. Prove that \(F_1 + F_2 + F_3 + \cdots + F_n = F_{n+2} - 1\).
3. Prove that for all integers \(n \ge 1\), \(8^n – 3^n\) is divisible by 5.
4. For all \(n \in \mathbb{N}\), prove that \(2^n \le 2^{n+1} - 2^{n-1} - 1\).