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

 Mar 13, 2023
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Let's represent the first fraction that Levans writes as `n/(n-1)`, where `n` is an integer. Levans then writes several more fractions by increasing both the numerator and denominator by 1, so the second fraction is `(n+1)/n`, the third is `(n+2)/(n+1)`, and so on. 

After writing 20 fractions, the product of all the fractions is:

```
n/(n-1) * (n+1)/n * (n+2)/(n+1) * ... * (n+19)/(n+18)
```

Simplifying this expression, we see that most of the terms cancel out:

```
n/(n-1) * (n+1)/n * (n+2)/(n+1) * ... * (n+19)/(n+18) = (n+19)/(n-1)
```

We know that the product of all the fractions is equal to 3, so:

```
(n+19)/(n-1) = 3
```

Multiplying both sides by `n-1`, we get:

```
n+19 = 3(n-1)
```

Expanding the right side, we get:

```
n+19 = 3n-3
```

Subtracting `n` from both sides, we get:

```
19 = 2n-3
```

Adding 3 to both sides, we get:

```
22 = 2n
```

Dividing both sides by 2, we get:

```
n = 11
```

Therefore, the first fraction that Levans writes is `11/10`.

 Mar 13, 2023

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