proof by induction

Question

Use the method of proof by induction to show that (1+x+x² +. . .+x^n-1)(1−x) = (1−xⁿ), for all positive integers n

Answer 1

The proof proceeds by induction.

 

When n = 1 we have (1)(1-x) = (1-x). Clearly true. (Note also 1 = x⁰ = x^n-1, so we can start our induction as we did, at n=1)

 

Now suppose it is true for n. We endeavor to show it is true for n+1.

 

i.e. Suppose n > 1, and it is established that (1+x+x² +. . .+x^n-1)(1−x) = (1-xⁿ)   (this is the inductive hypothesis). Then

 

(1 + x + x² + ... + x^n-1 + xⁿ)(1-x)

= (1 + x + x² + ... + x^n-1)(1-x) + xⁿ(1-x)  by the distributive law

= (1-xⁿ) + xⁿ(1-x)                                     where the left summand comes from the inductive hypothesis

= (1 - xⁿ⁺¹)                                               by distributing and cancelling.