+140 Simplify \(\frac{x^2+2x^4+3x^6+\dots+1005x^{2010}}{2x+4x^3+6x^5+\dots+2010x^{2009}}\)
+2666 Factor out x from the numerator of the fraction: \(x(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})\)
Now, factor out 2 from the denominator: \(2(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})\)
So we have \({x(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009}) \over 2(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})} = \color{brown}\boxed{x \over 2}\)
+288 Suppose that the expression equals y.
Dividing by x gives us, (x^2 + 2x^4 +... + 1005x^(2010))/(2x^2 + 4x^4 +... + 2010x^(2010)). This gives us a value of 1/2.
1/2 divided by x gives us \( \frac{1}{2x} \)
+2666 Factor out x from the numerator of the fraction: \(x(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})\)
Now, factor out 2 from the denominator: \(2(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})\)
So we have \({x(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009}) \over 2(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})} = \color{brown}\boxed{x \over 2}\)
