A full induction argument

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

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Before you start

Proof by induction. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

For integers n ≥ 0, consider $4^{n}$ − 1.

Part a

Evaluate the expression at n = 0.

Part b

Prove by induction that the expression is divisible by 3.

Part c

Explain why checking the first ten cases does not replace the proof.

Model solution and review criteria

Part a

$a^{0}$ − 1 = 0, which is divisible by every positive integer.

Check the stated starting index.

Part b

Assume $4^{k}$ −1 = (3)m for an integer m at an arbitrary k ≥ 0. Then 4^(k+1)−1 = 4( $4^{k}$ −1)+(3) = (3)(4m+1), an integer multiple. Together with the base case, induction proves the claim.

State the hypothesis at an arbitrary k.
Derive the next case using that hypothesis.
Identify the integer multiplier and conclude for all stated n.

Part c

A finite list does not establish every integer n. The implication from an arbitrary k to k+1 is what propagates the base case indefinitely.

Distinguish finite evidence from a universal argument.

Compare your own reasoning. No automatic examination marks are awarded.