Terminating Fractions

Algebra Level pending

1 2 ⋅ 2 3 ⋅ 3 4 ⋅ 4 5 ⋅ … ⋅ x = 1 100 \large \frac12 \cdot\frac23 \cdot \frac34 \cdot\frac45 \cdot \ldots \cdot x = \frac1{100}

Above shows the product of x x fractions such that its product equals to 1 100 \frac{1}{100} . If each of these x x fractions have their numerator is 1 less than its denominator, find the value of x x .

No Solution 99 101 100

Raunak Bele by Raunak Bele , 21, India

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2 solutions

Harshit Singhania
Jun 19, 2015

Let's take an example 1 2 × 2 3 × 3 4 × 4 5 \frac {1}{2}×\frac {2}{3}×\frac {3}{4}×\frac {4}{5} is equal to 1 5 \frac {1}{5} and this sequence had 4 terms so it is observed that for such a sequence of x terms the result is equal to 1 x + 1 \frac{1}{x+1} hence 1 100 = 1 x + 1 \frac {1}{100} = \frac {1}{x+1} which gives x = 99

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Raunak Bele
Jun 19, 2015

Yeah harshit that's correct

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