
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 19th, 2017, 07:33 AM  #1 
Member Joined: Sep 2014 From: Sweden Posts: 80 Thanks: 0  Induction: Prove that 2^n > (n+2)^2 when n > 6
Prove that $\displaystyle \displaystyle 2^n > (n+2)^2 \ when \ n > 6$ Base case: $\displaystyle \displaystyle P(7): 2^7 > (7+2)^2 = 128 > 81 \ true \\\\ $ Induction Hypothesis:\\ $\displaystyle \displaystyle P(n) \Rightarrow P(n+1)\\\\ \displaystyle Knows: \ 2^n > (n+2)^2\\\\ \displaystyle Wants: \ 2^{n+1}>(n+1+2)^2 = 2^{n+1}>(n+3)^2\\\\ $ Add 5 + 2n to both sides $\displaystyle \displaystyle 5+2n+2^n > n^2+4n+4+5+2n\\\\ \displaystyle 5+2n+2^n > n^2+6n+9\\\\ \displaystyle 5+2n+2^n>(n+3)^2\\\\ \displaystyle 5+2n+(n+2)^2>(n+3)^2\\\\ \displaystyle 5+2n+n^2+4n+4>(n+3)^2\\\\ \displaystyle n^2+6n+9>(n+3)^2\\\\ \displaystyle (n+3)^2>(n+3)^2\\\\ $ This would have worked fine if it was an equal sign $\displaystyle (\displaystyle = or\leq or\geq) $, but it is not. We all know that the same number can't be greater than the same number, it's equal. Is there any great videos or tutorials that explain mathematical induction basics really good? Or an website with induction assignments and so you can see the correct answer/answers? Last edited by DecoratorFawn82; October 19th, 2017 at 07:37 AM. 
October 19th, 2017, 08:43 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,059 Thanks: 1619 
Instead of adding $5 + 2n$ to each side, double each side: $2^{n+1} = 2(2^n) > 2(n + 2)^2 = (n + 3)^2 + (n + 3)(n  1) + 2 > (n + 3)^2$ 
October 19th, 2017, 09:27 AM  #3 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,797 Thanks: 715 Math Focus: Wibbly wobbly timeywimey stuff.  
October 19th, 2017, 10:16 AM  #4 
Member Joined: Sep 2014 From: Sweden Posts: 80 Thanks: 0 
I still don't see how we know that $\displaystyle \displaystyle (n+3)^2+(n+3)(n1)+2>(n+3)^2$ though? $\displaystyle \displaystyle (n+3)(n1)+2>0$ That's why right? Last edited by DecoratorFawn82; October 19th, 2017 at 10:33 AM. 
October 19th, 2017, 02:57 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 19,059 Thanks: 1619 
Yes, that's why.


Tags 
>, >, induction, prove 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Is it possible to prove it by induction?  davedave  Algebra  4  October 15th, 2013 10:50 PM 
Prove by induction  Sissy  Applied Math  2  November 25th, 2010 09:55 AM 
Prove by induction  MathematicallyObtuse  Calculus  2  November 22nd, 2010 02:46 PM 
Prove by induction  remeday86  Applied Math  2  July 20th, 2010 05:01 PM 
Prove by Induction  jrklx250s  Real Analysis  1  December 31st, 1969 04:00 PM 