# Thread: difference of b to base 2010 and b to base 2009

1. ## difference of b to base 2010 and b to base 2009

thanks a lot

2. $\displaystyle b_{2010}=\frac{1-b_{2009}}{1+b_{2009}}, \ b_{2009}=\frac{1-b_{2008}}{1+b_{2008}}$

$\displaystyle \frac{1-b_{2009}}{1+b_{2009}}-\frac{1-b_{2008}}{1+b_{2008}}=\frac{(1-b_{2009})(1+b_{2008})-(1-b_{2008})(1+b_{2009})}{(1+b_{2009})(1+b_{2008})}=\ frac{2b_{2008}-2b_{2009}}{(1+b_{2009})(1+b_{2008})}$

3. The first thing I would do is calculate a few values:
$b_1= \frac{1}{3}$
$b_2= \frac{1- \frac{1}{3}}{1+ \frac{1}{3}}= \frac{2}{3}\frac{3}{4}= \frac{1}{2}$
$b_3= \frac{1- \frac{1}{2}}{1+ \frac{1}{2}}= \frac{1}{2}\frac{2}{3}= \frac{1}{3}$
Of course then
$b_4= \frac{1- \frac{1}{3}}{1+ \frac{1}{3}}= \frac{2}{3}\frac{3}{4}= \frac{1}{2}$
and
$b_5= \frac{1- \frac{1}{2}}{1+ \frac{1}{2}}= \frac{1}{2}\frac{2}{3}= \frac{1}{3}$
Get the point?

4. I wasn't obtaining a pattern when I did it. I guess addition and subtraction have gotten tougher.

5. thanks for giving me hits guys! Long Live! God Bless.

6. Of course you really need to prove that $b_n= \frac{1}{3}$ for n odd and $\frac{1}{2}$ for n even, but that is easy using induction.