# proof by induction with inequalities

• Oct 11th 2012, 04:33 AM
narjsingh
proof by induction with inequalities
question is prove by induction on m that m^3 <= 2^m for m=>10
thats supposed to be m cube less than or equal to 2 to the power n for m greater than or equal to 10

P(K)= k^3 less than or equal to 2^k
P(K+1)= (k+1)^3 less than or equal to 2^k+1

2k^3 less than or equal to 2^k+1

(k+1)^3 less than or equal to 2k^3
now im not sure to do next
• Oct 11th 2012, 04:57 AM
emakarov
Re: proof by induction with inequalities
\begin{align*} (k+1)^3&=k^3+3k^2+3k+1\\ &\le k^3+3k^2+3k^2+k^2&& \text{since }k\ge3 \\&=k^3+7k^2\\&\le k^3+k^3&& \mbox{since }k\ge7\\&\le2^k+2^k&& \text{by induction hypothesis}\\&=2^{k+1}\end{align*}

A couple of remarks. For each claim it should be clear whether you are assuming it, trying to prove it or something else. Your proof is just a series of statements, and the role of those statements is not clear. Write comments when necessary, such as "We assume ...," "We need to prove ...," "It follows from ... that ..."

Mathematics is case-sensitive, so K and k may denote different variables.

Edit: Don't forget the base case.
• Oct 12th 2012, 12:10 PM
narjsingh
Re: proof by induction with inequalities
Quote:

Originally Posted by emakarov
\begin{align*} (k+1)^3&=k^3+3k^2+3k+1\\ &\le k^3+3k^2+3k^2+k^2&& \text{since }k\ge3 \\&=k^3+7k^2\\&\le k^3+k^3&& \mbox{since }k\ge7\\&\le2^k+2^k&& \text{by induction hypothesis}\\&=2^{k+1}\end{align*}

thanks but i dont understand where the second line has come from, how did you get k^3+3k^2+3k^2+k^2
• Oct 12th 2012, 12:30 PM
Plato
Re: proof by induction with inequalities
Quote:

Originally Posted by narjsingh
thanks but i dont understand where the second line has come from, how did you get k^3+3k^2+3k^2+k^2

$1\le k^2$
• Oct 12th 2012, 12:39 PM
narjsingh
Re: proof by induction with inequalities
Quote:

Originally Posted by Plato
$1\le k^2$

im really really sorry i still dont get that im a bsc econ student :(
• Oct 12th 2012, 12:51 PM
Plato
Re: proof by induction with inequalities
Quote:

Originally Posted by narjsingh
im really really sorry i still dont get that im a bsc econ student :(

Because $k\le k^2~\&~1\le k^2$ we get $3k+1\le 3k^2+k^2$

$k^3+3k^2+3k+1\le k^3+3k^2+3k^2+k^2$
• Oct 12th 2012, 12:59 PM
narjsingh
Re: proof by induction with inequalities
Quote:

Originally Posted by Plato
Because $k\le k^2~\&~1\le k^2$ we get $3k+1\le 3k^2+k^2$

$k^3+3k^2+3k+1\le k^3+3k^2+3k^2+k^2$

thank you so much! i am really getting it! one last hing how did you get k is greater than or equal to 7? :)
• Oct 12th 2012, 01:08 PM
Plato
Re: proof by induction with inequalities
Quote:

Originally Posted by narjsingh
thank you so much! i am really getting it! one last hing how did you get k is greater than or equal to 7? :)

$3+3+1=7$ so $3k^2+3k^2+k^2=7k^2$