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# Sequences and Series

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I have no idea how to do this stuff so when answering, please give me an answer and the reasoning behing it.

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 12, 2023

#1
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Let the first fraction that Levans wrote be n/(n - 1). Then the other 19 fractions are (n + 1)/n​,(n + 2)/(n + 1)​,…,(n + 19)/(n + 18)​. The product of all 20 fractions is then

\begin{align*} \frac{n}{n-1} \cdot \frac{n+1}{n} \cdot \frac{n+2}{n+1} \cdots \frac{n+19}{n+18} &= \frac{n(n+1)(n+2) \cdots (n+19)}{(n-1)n(n+1) \cdots (n+18)} \ &= \frac{n+19}{n-1}. \end{align*}

We are given that this product equals 3, so (n + 19)/(n - 1)​=3. Solving for n, we find n=10. Therefore, the first fraction that Levans wrote is n/(n - 1) = 10/9.

May 13, 2023
#2
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[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 ==2n

n ==20 / 2==10

So, the first fraction is: [n + 1] / n==[10 + 1] / 10 ==11 / 10

Check: productfor(n, 1, 20, (10 + n) / (9 + n))==3

May 13, 2023
#3
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The first fraction that Levans wrote has a numerator that is 1 greater than the denominator. This means that the numerator is n and the denominator is n−1. The next fraction that Levans wrote has a numerator of n+1 and a denominator of n. This pattern continues for 20 fractions.

The product of all 20 fractions is 3. This means that the numerators and denominators of the fractions must cancel each other out, leaving only 3. The only way for this to happen is if the numerators and denominators are consecutive integers.

The first fraction that Levans wrote has a numerator of n and a denominator of n−1. The next fraction has a numerator of n+1 and a denominator of n. This means that the numerators of the fractions are n,n+1,n+2,...,n+19. The denominators of the fractions are n−1,n,n+1,...,n+18.

The product of the numerators is n(n+1)(n+2)...(n+19). The product of the denominators is (n−1)n(n+1)...(n+18). The numerators and denominators cancel each other out, leaving only n+19.

The value of the first fraction that Levans wrote is n−1n​. The value of the product of all 20 fractions is n−1n+19​. The value of the product of all 20 fractions is 3. This means that (n+19)/(n-1)​=3. Solving for n, we find that n=16.

Therefore, the value of the first fraction that Levans wrote is 16/15.

May 13, 2023
#4
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Check: (16 / 15, 17 / 16, 18 / 17, 19 / 18, 20 / 19, 21 / 20, 22 / 21, 23 / 22, 24 / 23, 25 / 24, 26 / 25, 27 / 26, 28 / 27, 29 / 28, 30 / 29, 31 / 30, 32 / 31, 33 / 32, 34 / 33, 35 / 34)==7/3 - your solution

Check: (11 / 10, 12 / 11, 13 / 12, 14 / 13, 15 / 14, 16 / 15, 17 / 16, 18 / 17, 19 / 18, 20 / 19, 21 / 20, 22 / 21, 23 / 22, 24 / 23, 25 / 24, 26 / 25, 27 / 26, 28 / 27, 29 / 28, 30 / 29)==3 - the one above you.

May 13, 2023