What You Need to Know

Step 1 (Base case): prove the statement is true for n = 1 (or n = 0)

Step 2 (Inductive step): assume the statement is true for n = k (inductive hypothesis)

Step 3: prove the statement is true for n = k + 1 using the inductive hypothesis

If both steps hold, the statement is true for all natural numbers

Induction is used to prove summation formulas, divisibility, and inequalities

Key Vocabulary

A proof technique for proving statements about all natural numbers

The initial step in a proof by induction, usually proving the statement for n = 1

The assumption that the statement is true for n = k

Proving the statement for n = k+1 given it holds for n = k

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

In a proof by induction, the base case for a statement about natural numbers is typically:

Proving it for n = 1 Proving it for all n Proving it for n = k Proving it for n = k + 1

Question 2

The inductive hypothesis assumes:

The statement is true for all n The statement is true for n = k The statement is true for n = 1 The statement is false for n = k

Question 3

If the base case holds and the inductive step is proven, the statement is:

True for n = 1 only True for n = 2 only True for all natural numbers Possibly true for some numbers