+570 Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 2 greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator of the previous fraction by 1. He then multiplies all his fractions together. He has 3 fractions, and their product equals 10. What is the value of the first fraction he wrote?
+15000 What is the value of the first fraction he wrote?
\(\frac{x}{x+2}\cdot \frac{x+1}{x+3}\cdot\frac{x+2}{x+4}=10\\ \frac{x^2+x}{x^2+7x+12}=10\\x^2+x=10x^2+70x+120\\ 9x^2+69x+120=0\\ 3x^2+23x+40=0\\\)
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\\ x = {-23 \pm \sqrt{23^2-4\cdot 3\cdot 40} \over 2\cdot 3}\\ \color{blue}x\in\{-2\frac{2}{3},-5\} \)
\(\frac{x}{x+2}\cdot \frac{x+1}{x+3}\cdot\frac{x+2}{x+4}=10\\ x=-5\\ \frac{-5}{-3}\cdot \frac{-4}{-2}\cdot\frac{-3}{-1}=10\\\)
The value of the first fraction he wrote is \(\color{blue}\frac{5}{3}.\)
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