mathschallenge.net

Problem

Prove that n must be odd for (1 + ¹/₂)(1 + ¹/₃)(1 + ¹/₄)...(1 + ¹/_n) to be integer?

Solution

Begin by using the result:

n + 1 n

= 1 +

1 n

So that we can write the product in the following way and cancel denominators,

1 +

1 2

1 +

1 3

1 +

1 4

...

1 +

1 n

=

3 2

4 3

5 4

5

...

n 1

n n 1

n + 1 n

=

n + 1 2

Hence it will be integer when n is odd.