Proof by Induction

Front

What is the principle of mathematical induction?

Back

A method of proving that a statement is true for all natural numbers by proving it for the first number and showing that if it's true for an arbitrary number k, it must also be true for k+1.

Front

What are the two main steps in a proof by induction?

Back

1. Prove the base case (usually n=1). 2. Prove the inductive step (if true for n=k, then true for n=k+1).

Front

What is the base case in mathematical induction?

Back

The base case is the initial step where the statement is proven true for the first natural number, typically n=1.

Front

What is the inductive hypothesis?

Back

The assumption that the statement is true for n=k in order to prove it for n=k+1.

Front

What does it mean to prove a statement for n=k+1?

Back

It means to show that if the statement holds for n=k, then it also holds for the next natural number, n=k+1.

Front

What is the formula for the sum of the first n odd numbers?

Back

1 + 3 + 5 + ... + (2n - 1) = n^2.

Front

How do you express S(n) for the sum of the first n odd numbers?

Back

S(n) = 2n - 1.

Front

What is the result of S(k) when k is a natural number?

Back

S(k) = 2k - 1.

Front

What is the result of S(k+1) when k is a natural number?

Back

S(k+1) = 2(k+1) - 1 = 2k + 1.

Front

What is the significance of the induction assumption in a proof?

Back

It allows us to assume the statement is true for n=k to prove it for n=k+1.