Solve $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right)=2010x^{2009}$

Question

Solve for $x$
$\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right)=2010x^{2009}$

solution should be by hand

Answer 1

First it is clear that $x \geq 0$.

By AM-GM inequality we have:

$$x^{2010}+1 \geq 2 x^{1005}$$ $$1+x^2+x^4+x^6+.......+x^{2008} \geq 1005 \sqrt[1005]{1x^2...x^{2008}}=1005 x^{\frac{2+4+6+...+2008}{1005}}=1005x^{1004}$$

Multiplying you get $$\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right) \geq2010x^{2009}$$

You get equality if and only if you have equality in the above. Thus

$$x^{2010} =1$$ and $$1=x^2=x^4=...=x^{2008}$$

Answer 2

Note that in your second term on the left hand side, you have

$$\sum_{n=0}^{1004} (x^2)^n.$$

This is a geometric series which gives $\frac{1-(x^2)^{1005}}{1-x^2}$. See here. Do you see where to go from here?