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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?

 Mar 26, 2023
 #1
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Let x be the denominator of the first fraction that Levans writes. Then, the numerator of this fraction is x + 1, as given in the problem statement.

For each subsequent fraction that he writes, both the numerator and the denominator are increased by 1. So, the second fraction has a numerator of (x + 2) and a denominator of (x + 3), the third fraction has a numerator of (x + 3) and a denominator of (x + 4), and so on.

Levans writes a total of 20 fractions, and their product is 3. We can express the product of all 20 fractions as a single fraction with numerator equal to the product of all the numerators and denominator equal to the product of all the denominators:

(x + 1)(x + 2)(x + 3)...(x + 20) ---------------------------------- (x + 2)(x + 3)(x + 4)...(x + 21)

We know that the product of all these fractions is equal to 3. So, we have the equation:

(x + 1)(x + 2)(x + 3)...(x + 20) ---------------------------------- = 3 (x + 2)(x + 3)(x + 4)...(x + 21)

To solve for x, we can start by simplifying both the numerator and the denominator by canceling out any common factors. For example, (x + 2)/(x + 2) = 1, so we can cancel out all terms of the form (x + n)/(x + n) for n = 2 to 21:

(x + 1)/(x + 21) = 3/840

We can then cross-multiply to get:

840(x + 1) = 3(x + 21)

Simplifying this equation gives:

837x = 597 x = 597/837

Therefore, the value of the first fraction that he wrote is:

(x + 1)/x = (597/837 + 1)/(597/837) = 1810/597

Hence, the value of the first fraction that Levans wrote is 1810/597.

 Mar 26, 2023
 #2
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n / [n - 1] * [n + 1] / n * [n + 2] /[n + 1] * [n + 3] / [n + 2] *..........* [n + 19] / [n + 18]==3, solve for n

 

They all cancel out except:

 

[n + 19] / [n - 1] ==3

[n  + 19] ==[3n  -  3]

19 + 3 ==3n - n

22 == 2n

n==22 / 2==11 

 

Therefore, the first fraction was: 11 / 10

Check: 11 / 10  *  12 / 11  *  13 / 12  *..........* 30 / 29 ==3

 Mar 27, 2023

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