June 1st, 2012, 07:22 PM  #1 
Newbie Joined: Jun 2012 Posts: 16 Thanks: 0  Induction
Hi Try to Figure out the next step in this problem, I have the answer but I don't know how to arrive at it 1x2+2x3+3x4+...+n(n+1)=(n(n+1)(n+2))/3 n=1, 1x2=1(1+1)(1+2) 2=6/3 2=2 thus the statement holds for n=1 Assume true for n=k 1x2+2x3+3x4+...+k(k+1)=(k(k+1)(k+2))/3 Prove true for n=k+1 1x2+2x3+3x4+...+k(k+1)+(k+1)(k+2)=((k+1)(k+2)(k+3) )/3 ((k(k+1)(k+2))/3) +(k+1)(k+2)=((k+1)(k+2)(k+3))/3 ((k(k+1)(k+2))/3) +(3(k+1)(k+2)/3)=((k+1)(k+2)(k+3))/3 This is as far as I can go, and I'm not sure if the steps I have taken are correct (I think I forgot something on the right side) can someone please explain how to arrive at ((k+1)(k+2)(k+3))/3=((k+1)(k+2)(k+3))/3 Then more detailed the explanation the better Thankyou 
June 1st, 2012, 08:05 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs  Re: Induction
You showed the base case is true, so state your induction hypothesis : Add to both sides: Factor on the right side: We have derived from thereby completing the proof by induction. 
June 1st, 2012, 08:09 PM  #3  
Member Joined: Jul 2010 Posts: 42 Thanks: 0  Re: Induction Quote:
 
June 1st, 2012, 08:59 PM  #4  
Newbie Joined: Jun 2012 Posts: 16 Thanks: 0  Re: Induction Quote:
 
June 1st, 2012, 09:07 PM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs  Re: Induction
It can be shown by factoring very similarly to what I did in my post above: 
June 1st, 2012, 09:21 PM  #6 
Newbie Joined: Jun 2012 Posts: 16 Thanks: 0  Re: Induction
I get IT now!!! A big thank you to both of you!!!

June 1st, 2012, 09:23 PM  #7 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs  Re: Induction
Glad to help and welcome to the forum! 
June 1st, 2012, 09:48 PM  #8 
Newbie Joined: Jun 2012 Posts: 16 Thanks: 0  Re: Induction 
June 1st, 2012, 09:56 PM  #9  
Newbie Joined: Jun 2012 Posts: 16 Thanks: 0  Re: Induction Quote:
Prove by induction the worked example for this exercise is assume n=k is true then n=k+1 must also be true why in the last step did he add to the right hand side Shouldn't it be:  
June 1st, 2012, 10:06 PM  #10 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs  Re: Induction
Your last statement is (almost) where we want to wind up. After having demonstrated the base case is true, we state the induction hypothesis Now, we must arrive at algebraically, so we add the same thing to both sides, in this case : We have derived from thereby completing the proof by induction. You see, with induction, we must demonstrate that we can, using legal algebraic operations, get from to which means it must be true for any k. By the way, once you have made 20 or more posts, you will be able to edit your posts. It is an unfortunate antispam measure we've had to adopt. We apologize for the inconvenience. 

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