Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 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 . He then multiplies all his fractions together. He has fractions, and their product equals . What is the value of the first fraction he wrote?
Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is $1$ 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 $20$ fractions, and their product equals $3$. What is the value of the first fraction he wrote?
Let a and b be the numerator and denominator of the first fraction. We know that a=b+1. The product of the fractions is: \begin{align*} \frac{a}{b}\cdot\frac{a+1}{b+1}\cdot\frac{a+2}{b+2}\cdots\frac{a+19}{b+19}&=\frac{a^{20}}{b^{20}}\ &=\frac{(b+1)^{20}}{b^{20}} \end{align*}We are given that this product equals 3, so [(b+1)^{20}=3b^{20}.]We can factor out a b20 from both sides to get [b^{20}(b+1)=3b^{20}.]This simplifies to b+1=3, so b=2. Then a=b+1=3, so the first fraction is 2/3.
[n + 1] / n * [n + 2] / [n + 1] * [n + 3] / [n + 2] *..............* [n + 20] / [n + 19] ==3
All cancel out except: [n + 20] / n ==3
[n + 20] ==3n
20 =3n - n
20 =2n
n ==10 and the first fraction is: [10 + 1] / 10 ==11 / 10 - this is the first fraction.
Check: 11/10 * 12/11 * 13/12 * 14/13 *............* 30 /29 ==3
