Mathematical Induction

Prove a statement for every whole number at once.

12 problems · about 22 minutes · TEKS P.5.E

NameDate

Figure — use these to answer the problems

The sum 1 + 2 + ... + n equals n(n+1)/2, a quadratic. Induction is what proves that formula holds forever.

y = 0.5x² + 0.5x + 0

a0.5

b0.5

Figure 1

Straightforward practice. Get the method working first.

5 problems

n(n+1)/2 at n = 8?
Answer
2
Why is a base case necessary?
- Without a true first case, the chain of implications never starts.
- It makes the proof look complete.
n(n+1)/2 at n = 6?
Answer
n(n+1)/2 at n = 10?
Answer
The sum of the first n odd numbers is n². What is it for n = 5?
Answer

The same ideas with more to keep track of.

3 problems

How many parts does a proof by induction have?
Answer
In the inductive step, which case do you assume? Enter the value as a number of steps past k, so 0 means k itself.
Answer
The sum of the first n odd numbers at n = 8?
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Put a proof by induction in order.
Write 1 to 4 in the boxes to put these in order.
- State the formula to be proved.
- Verify the base case, usually n = 1.
- Assume the formula holds for n = k.
- Derive the formula for n = k + 1.
n(n+1)/2 at n = 100?
Answer
The Base
For 1 + 2 + ... + n = n(n+1)/2, what does the right side give when n = 4?
Answer
The Missing Part
A proof verifies n = 1 but never proves the step from k to k + 1. Is it complete? Enter 1 for yes, 0 for no.
Answer