Fun Product

Algebra Level 2

Evaluate the following expression:

( 1 + 1 1 ) ( 1 + 1 2 ) ( 1 + 1 3 ) ( 1 + 1 4 ) ⋯ ( 1 + 1 2018 ) \sqrt{\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right) \cdots \left(1+\frac{1}{2018}\right)}

2018 \sqrt{2018} 2019 \sqrt{2019} 2017 \sqrt{2017} 45

Aidan Poor by Aidan Poor , 18, USA

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1 solution

Parth Sankhe
Oct 23, 2018

Convert the sum in each term into a fraction to give:

2 1 ⋅ 3 2 ⋅ 4 3 ⋅ 5 4 . . . . . 2018 2017 ⋅ 2019 2018 \sqrt {\frac {2}{1} \cdot \frac {3}{2} \cdot \frac {4}{3} \cdot \frac {5}{4}.....\frac {2018}{2017} \cdot \frac {2019}{2018}}

All these terms cancel out to give 2019 \sqrt {2019} as the answer.

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