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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?

May 14, 2022

#1
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Let (n + 1)/n be the first fraction.

$$\dfrac{n + 1}n \cdot \dfrac{n + 2}{n + 1} \cdots \dfrac{n+20}{n + 19} = 3$$

Note that the left-hand side is just a telescoping product, so $$\dfrac{n + 20}n = 3$$.

Please solve the equation for n and then find the value of (n + 1)/n on your own.

May 14, 2022
#3
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Notice that for any 2 consecutive terms in the product, the numerator of the first fraction always cross out with the denominator of the second fraction. That leaves us with the numerator of the last fraction, divided by the denominator of the first fraction, which is (n + 20)/n. If you are attempting the solve the equation, you should try to multiply both sides by n so that the equation becomes linear.

MaxWong  May 15, 2022