Mathematical Induction

The Principle of Mathematical Induction Let S n be a statement involving the positive integer n. If 1.S 1 is true, and 2.the truth of the statement S k implies the truth of the statement S k+1, for every positive integer k, then the statement S n is true for all positive integers n.

The Steps in a Proof by Mathematical Induction Let S n be a statement involving the positive integer n. To prove that S n is true for all positive integers n requires two steps. STEP 1 Show that S 1 is true STEP 2 Show that if S k is assumed to be true, then S k+1 is also true, for every positive integer k.

Example For the given statement S n, write the three statements S 1, S k, and S k+1 Solution:

For the given statement S n, write the three statements S 1, S k, and S k+1. Text Example Solution We begin with Write S 1 by taking the first term on the left and replacing n with 1 on the right.

Solution Write S k by taking the sum of the first k terms on the left and replacing n with k on the right. Write S k+1 by taking the sum of the first k + 1 terms on the left and replacing n with k + 1 on the right. Simplify on the right. Text Example cont.

Example Use mathematical induction to prove that Solution: Step 1:

Example cont. Use mathematical induction to prove that Solution: Step 2:

Mathematical Induction