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 December 7th, 2010, 05:12 PM #1 Member   Joined: Oct 2009 Posts: 98 Thanks: 0 Is this a sequence? Series? There's a challenge problem on my homework and I have no real idea how to start it. Maybe if you guys could give me a hint, I could finish it. Here is a picture of it. I wasn't sure how to LaTeX this one. Uploaded with ImageShack.us I don't necessarily need someone to do this entire problem, unless you can explain important steps, because I want to understand this. I just don't know where to start because the book we have has nothing close to something like this in the example and practice problems. Thanks in advance for your help.
 December 7th, 2010, 07:14 PM #2 Senior Member   Joined: Nov 2010 From: Staten Island, NY Posts: 152 Thanks: 0 Re: Is this a sequence? Series? I think induction should work. Write $33...34= 30...00 + 3...34$, then FOIL. The first, outer and inner are easy to compute, and the last is done by the inductive assumption. Adding these up is easy and should give the left hand side.
 December 7th, 2010, 09:28 PM #3 Member   Joined: Oct 2009 Posts: 98 Thanks: 0 Re: Is this a sequence? Series? Ok, so I really don't understand what is going on with this problem here. You said to foil the right side, but I don't know how to do that when it's 333333333. I know the foil method, but I just don't know what properties to use with this problem. Keep in mind, we have done nothing like this in class and I would like an explanation of this problem more than just the answer. Thanks for your time!
 December 8th, 2010, 01:31 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,105 Thanks: 1907 3.  Prove that $\underbrace{111\,...\,1}_k \underbrace{55\,...\,5}_{k-1}\6\,=\,\left(\underbrace{33\,...\,3}_{k-1}\,4\right)^2$ The LHS is $(10^{2k}-1)/9\,+\,(4/9)(10^k-1)\,+\,1$ and the RHS is $((10^k-1)/3\,+\,1)^{\small2}.$ It's easy to prove that those two expressions are equal.
 December 8th, 2010, 07:29 AM #5 Senior Member   Joined: Nov 2010 From: Staten Island, NY Posts: 152 Thanks: 0 Re: Is this a sequence? Series? Let me give some more details: $(33...34)^2= (30...00 + 3...34)^2 = 30...00^2 + 2(30...00)(3...34)+3...34^2 = 300...0000+ 20...0040...00+111...155...56=11...15...56$ You should be a bit more detailed by noting how many of each number appears at each stage. As a specific example here I will do k=3 in detail: $(334)^2=(300+34)^2 = 90000+20400+1156=111556$ The inductive hypothesis was used in $34^2=1156$.
December 8th, 2010, 08:38 AM   #6
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Re: Is this a sequence? Series?

Quote:
 Originally Posted by suomik1988 ...but I don't know how to do that when it's 333333333. ...
All of the preceding advice is correct - just thought I'd throw in my \$0.02
...
"333333333" is really
33333333334, which you can write as
30000000000 +
3333333334
But you know what 3333333334 "is" (so to speak) from the inductive hypothesis.
So you have
30000000000² + 2*30000000000*3333333334 + 3333333334² after foiling...

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