Easy one!

Algebra Level 2

( 1 − 1 2 ) ( 1 − 1 3 ) ( 1 − 1 4 ) … ( 1 − 1 999 ) ( 1 − 1 1000 ) \left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\ldots\left(1-\frac{1}{999}\right)\left(1-\frac{1}{1000}\right)

If the value of the expression above can be expressed as a b \frac ab for coprime positive integers a a and b b , find the value of a + b a+b .


The answer is 1001.

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Adam Phúc Nguyễn by Adam Phúc Nguyễn , 20, Vietnam

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2 solutions

( 1 − 1 2 ) ( 1 − 1 3 ) ( 1 − 1 4 ) … ( 1 − 1 999 ) ( 1 − 1 1000 ) \left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\ldots\left(1-\frac{1}{999}\right)\left(1-\frac{1}{1000}\right) = 1 2 × 2 3 × 3 4 × … × 998 999 × 999 1000 = \frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\ldots\times\frac{998}{999}\times\frac{999}{1000} = 1 1000 =\boxed{\frac{1}{1000}}

9 Helpful 0 Interesting 0 Brilliant 0 Confused

Try this

Nihar Mahajan - 5 years, 8 months ago
Mehdi Balti
Oct 8, 2015

So simple just put in 1/n where, n=1000 :)

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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