What's the Difference Between the Inductive Hypothesis and the Inductive Step?

Two phrases appear side by side in nearly every induction proof, and they sound so similar that students routinely treat them as the same thing:

Inductive hypothesis: "Assume P(k) is true."
Inductive step: "If P(k) is true, then P(k+1) is true."

Are these two names for one thing? Is the hypothesis inside the step? Is the step after the hypothesis? The vocabulary is casual enough that textbooks blur the distinction, but the distinction is worth drawing cleanly — because once you see it, the structure of every induction proof becomes easier to write and easier to read.

The short answer

The inductive hypothesis is the assumption you make at the start of the inductive step. The inductive step is the entire proof of the conditional P(k) \Rightarrow P(k+1), including the hypothesis and the derivation that follows it.

Put differently: the inductive step is a little sub-proof. The inductive hypothesis is the first line of that sub-proof. The remaining lines do the algebra.

A two-column picture

Think of the inductive step as a small proof block with a top line (the assumption) and a body (the derivation):

Part of the inductive step	What it says	Role
Inductive hypothesis	"Assume P(k) holds for some arbitrary k \geq 1."	Opens the conditional sub-proof.
Derivation	Algebraic or logical work starting from P(k).	Builds toward P(k+1).
Conclusion	"Therefore P(k+1) holds."	Closes the conditional sub-proof.

All three of these together form the inductive step. The hypothesis is just the first piece.

Why this matters: when you write "by the inductive hypothesis," you are citing the assumption you made at the top of the step — not the step as a whole. When you announce "the inductive step goes through," you are claiming that the entire conditional P(k) \Rightarrow P(k+1) has been proved. The two are not interchangeable. Mixing them up usually produces a proof that says too little or too much.

A worked contrast

Take the familiar claim 1 + 2 + \dots + n = \frac{n(n+1)}{2}.

Inductive step (the whole block):

Let k \geq 1 be an arbitrary positive integer. Inductive hypothesis: assume 1 + 2 + \dots + k = \frac{k(k+1)}{2}. We show that 1 + 2 + \dots + k + (k+1) = \frac{(k+1)(k+2)}{2}.

Starting from the left-hand side:

1 + 2 + \dots + k + (k+1) = \frac{k(k+1)}{2} + (k+1) \quad \text{(by inductive hypothesis)}

= \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}.

This is P(k+1), so the inductive step is complete.

Notice two things in the layout:

The inductive hypothesis is the specific claim "1 + 2 + \dots + k = \frac{k(k+1)}{2}" — one equation, used once, as an assumption.
The inductive step is the entire block above: from "Let k \geq 1" down to "the inductive step is complete." It includes the hypothesis, the arithmetic, and the conclusion.

If I say "by the inductive hypothesis" mid-proof, I am referring back to the one equation in the first sentence — the fact I am allowed to use. If I say "the inductive step is complete," I am announcing that the whole sub-proof has succeeded.

A useful metaphor: ingredient vs recipe

A recipe contains several ingredients, plus a set of instructions that combines them. The inductive hypothesis is an ingredient — a specific statement you are putting into the recipe. The inductive step is the whole recipe, including that ingredient, the other ingredients (standard algebra rules), and the instructions that combine them into the finished dish (the conclusion P(k+1)).

A student who says "by the inductive step, \frac{k(k+1)}{2} + (k+1) = \frac{(k+1)(k+2)}{2}" is using the wrong word — that equation follows from algebra, not from the inductive step. A student who says "by the inductive hypothesis, P(k+1) holds" is also using the wrong word — the hypothesis only gives you P(k); reaching P(k+1) requires the full step, not just the hypothesis.

Why the confusion is easy

There are three reasons the two terms blur together:

Both contain the word "inductive." The prefix signals "related to induction," which is true of both, but does not distinguish between them.
The step's opening line is the hypothesis. Because every inductive step begins "Assume P(k)" (the hypothesis) as its first line, students often treat the opening line as the entirety of the step.
Casual usage in proofs. Writers sometimes say "by induction, \dots" or "by the inductive assumption, \dots" when they mean "by the inductive hypothesis." Everyone in the room understands, so the imprecision goes unchallenged.

The habit to cultivate is to use "inductive hypothesis" whenever you are citing the specific assumption P(k), and "inductive step" only when referring to the whole conditional proof. In practice, you will write the words "inductive hypothesis" dozens of times in a course — nearly every algebra step in the step invokes it — and you will write "the inductive step" only a few times, typically at the beginning and end.

A structural check for your own proofs

Here is a simple diagnostic to audit any induction proof you write:

Find the inductive hypothesis. It should be one sentence: "Assume P(k) holds" — specifically, an instance of the statement you are proving. If you cannot point to the one sentence that is the hypothesis, something is off.
Find the inductive step's boundary. The step starts with "Let k \geq 1 be arbitrary" (or equivalent) and ends with "so P(k+1) holds." Everything between those bookends is part of the step.
Verify that the hypothesis is used. If the derivation never cites the hypothesis — never writes "by the inductive hypothesis, \dots" — then your proof is actually a direct proof of P(k+1), not an induction. That might still be valid, but induction is the wrong label for it.
Verify that the conclusion matches P(k+1). The derivation should end with the statement you are trying to prove, specialised to n = k+1. If it ends with a different statement, the step has not reached the right destination.

Run this check on your last induction homework problem. You will often find that clarifying the two roles tightens the writing.

The one-line answer

The inductive hypothesis is the assumption "P(k) holds" — a single sentence. The inductive step is the entire conditional sub-proof that takes this assumption as its premise and derives P(k+1) as its conclusion. The hypothesis is the opening line of the step; the step is everything from the opening assumption to the closing conclusion. Using the two names precisely makes induction proofs easier to write and easier to read.