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# Hello!!!!!

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Find the value of $$6+\frac{1}{2+\frac{1}{6+\frac{1}{2+\frac{1}{6+\cdots}}}}$$  . Your answer will be of the form $$a+b\sqrt{c}$$  where no factor of c is a square. Find a+b+c .

This is for Melody.... Just ignore that dollar sign

Jun 29, 2017

#1
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Hi again Tertre :)

Let

$$x=6+\frac{1}{2+\frac{1}{6+\frac{1}{2+\frac{1}{6+\cdots}}}}\\ x=6+\frac{1}{2+\frac{1}{x}}\\ x=6+\frac{1}{\frac{2x+1}{x}}\\ x=6+\frac{x}{2x+1}\\ x(2x+1)=6(2x+1)+x\\ 2x^2+x=12x+6+x\\ 2x^2-12x-6=0\\ x^2-6x-3=0\\ x=\frac{6\pm\sqrt{36+12}}{2}\\ x=\frac{6\pm 4\sqrt{3}}{2}\\ x=3\pm 2\sqrt{3}\\ \text{But x cannot be negative so}\\ x=3+2\sqrt{3}\\~\\ 6+\frac{1}{2+\frac{1}{6+\frac{1}{2+\frac{1}{6+\cdots}}}}=3+2\sqrt{3}\\$$

a=3, b=2 c=3 so

a+b+c=8

I am wondering why you were asked for that sum?

Maybe to make a multiple choice question easier to mark ??

Or perhaps there is a quicker way to find that sum....

Jun 29, 2017
#2
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It's correct, thanks!

Jun 29, 2017
edited by tertre  Jun 29, 2017
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I hope I am not just doing your homework for you. :/

Melody  Jun 29, 2017