Simple Proof by Induction

Date: 08/27/99 at 06:10:52
From: Karim Bourouba
Subject: Proof by Induction

Hi,

Here's my question: Show by mathematical induction that the 
proposition below is true:

     If n >= 1 then 3n >= 1 + 2n

-Karim

Date: 08/28/99 at 02:39:17
From: Doctor Floor
Subject: Re: Proof by Induction

Dear Karim,

Thanks for sending your question.

The induction proof:

1. For n = 1 the statement is true, because then the statement reads

     3 >= 3.

2. Suppose that the statement is true for m, so 3m >= 1 + 2m.
   We consider m+1:

     3(m+1)   >=  1 + 2(m+1)
     (3m) + 3 >= (1 + 2m) + 2


   So the "3n" part grows more than the "1+2n" part, and we can 
   conclude that:

     3(m+1) >= 1 + 2(m+1).

   And the proof using mathematical induction stands.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/