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# Geometric series problem

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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 10 fractions, and their product equals 7/2. What is the value of the first fraction he wrote?

May 4, 2023

#1
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Let the first fraction be ba​. We are told that a>b and a=b+2. Therefore, we know that a=b+2=a/b​→b^22−ab−2b+4=0→(b−2)(b−1)=0. Therefore, b=1 or b=2. We know that b must be an integer, so b=2. Therefore, the first fraction is 4/2.​

May 4, 2023
#2
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a=productfor(n,11, 20, (n+2) / n)==7/2

First fraction == 13 / 11

Note: The quadratic equation is:  5d^2 - 35d - 220 = 0, which gives d==11 and [d + 2] / d ==13 / 11

May 5, 2023
#3
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The product of the fractions is 7/2​. The numerators and denominators of the fractions form an arithmetic sequence with a common difference of 1. The first numerator is 2 greater than the first denominator. Therefore, the first numerator is 3 and the first denominator is 1. The answer is 3/1.

May 5, 2023