mathe
- Thread starter mathe25
- Start date
Hello, mathe25!
\(\displaystyle \dfrac{x\cdot x^2\cdot x^3 \cdot x^4 \cdots x^n}{x\cdot x^3 \cdot x^5 \cdot x^7 \cdots x^{2n-1}}\)
. . \(\displaystyle \displaystyle =\;\frac{x}{x}\cdot\frac{x^2}{x^3}\cdot \frac{x^3}{x^5} \cdot\frac{x^4}{x^7} \cdots \frac{x^n}{x^{2n-1}}\)
. . \(\displaystyle \displaystyle =\;1\cdot\frac{1}{x}\cdot\frac{1}{x^2}\cdot\frac{1}{x^3} \cdots \frac{1}{x^{n-1}}\)
. . \(\displaystyle \displaystyle =\;\frac{1}{x^{[1+2+3+\cdots+(n-1)]}}\)
. . \(\displaystyle \displaystyle=\;\frac{1}{x^{\frac{n(n-1)}{2}}}\)
\(\displaystyle \dfrac{x\cdot x^2\cdot x^3 \cdot x^4 \cdots x^n}{x\cdot x^3 \cdot x^5 \cdot x^7 \cdots x^{2n-1}}\)
. . \(\displaystyle \displaystyle =\;\frac{x}{x}\cdot\frac{x^2}{x^3}\cdot \frac{x^3}{x^5} \cdot\frac{x^4}{x^7} \cdots \frac{x^n}{x^{2n-1}}\)
. . \(\displaystyle \displaystyle =\;1\cdot\frac{1}{x}\cdot\frac{1}{x^2}\cdot\frac{1}{x^3} \cdots \frac{1}{x^{n-1}}\)
. . \(\displaystyle \displaystyle =\;\frac{1}{x^{[1+2+3+\cdots+(n-1)]}}\)
. . \(\displaystyle \displaystyle=\;\frac{1}{x^{\frac{n(n-1)}{2}}}\)