Algebra and Trigonometry, 5th Edition

11.4 Mathematical Induction

• Prove infinite sequences of statements using mathematical induction.

In this section, we learn to prove a sequence of mathematical statements using a procedure called mathematical induction.

Proving Infinite Sequences of Statements

Infinite sequences of statements occur often in mathematics. In an infinite sequence of statements, there is a statement for each natural number. For example, consider the sequence of statements represented by the following:

“The sum of the first n positive odd integers is $n^2$,” or $1+3+5+\cdots +(2n-1)=n^2$.

Let’s think of this as $S(n)$, or $S_n$. Substituting natural numbers for n gives a sequence of statements. We list the first four:

• $$S_1:1=1^2;$$
• $$S_2:1+3=4=2^2;$$
• $$S_3:1+3+5=9=3^2;$$
• $$S_4:1+3+5+7=\mathrm{16\; ...}$$