Comp 283, 2025 Summer Session 1

Course Site for Comp 283

(Strong) Induction

Let’s Start With an Example

Example with Template

Recall the Fibonacci series…

We want to prove the following about the sum of the first \(n\) numbers of the series:

\(\forall n \in \mathbb{N}, F(0) + F(1) + F(2) + \ldots + F(n-1) + F(n) = F(n+2) - 1\).

S1. State the ‘for all’ statement that you want to prove:

S2. Say “we prove this by induction on” and state the induction parameter.

S3. Prove the base case(s).

S4. Write Induction Step.

S5. State the Induction Hypothesis (IH)

S6. State what you are going to prove about your specific value of \(x\) that was given to you in S4:

S7. Do the proof for the specific \(x\).

\begin{equation} \begin{array}{l l l} 1. & \forall 1 \leq k \leq n, {} \sum_{i=0}^k F(i) = F(k+2) - 1 & \textrm{IH}
2. & \textrm{Let } k = n - 1, \textrm{ then } \sum_{i=0}^{n-1} F(i) = F(n-1+2) - 1 & \textrm{Applied IH}
3. & \sum_{i=0}^{n-1} F(i) = F(n+1) - 1 & \textrm{Rewrote Line 2}
4. & \sum_{i=0}^{n-1} F(i) + F(n) = F(n+1) - 1 + F(n) & \textrm{Added } F(n) \textrm{ to both sides.}
5. & F(n+2) = F(n+1) + F(n) & \textrm{Fibonacci Def.}
6. & \sum_{i=0}^{n-1} F(i) + F(n) = F(n+2) - 1 & \textrm{Plugged 5. into 4.}
7. & \sum_{i=0}^n F(i) = \sum_{i=0}^{n-1} F(i) + F(n) & \textrm{Def. of Sum}
8. & \sum_{i=0}^{n} F(i) = F(n+2) - 1 & \textrm{Plugged 7. into 6.} \square \end{array} \end{equation
}

S8. Declare victory.

Tips for Proving Something by Induction (S7)