# reciprocals

• Dec 7th 2007, 06:51 PM
mr.toronto
reciprocals
Are there 21 different positive whole numbers such that the sum of their reciprocals is 1?

• Dec 7th 2007, 10:07 PM
Yes
Yes.

1/2+1/3 = 5/6
$1/2+1/3+\frac{1/2+1/3}{6}$ = 35/36
$1/2+1/3+\frac{1/2+1/3}{6}+\frac{\frac{1/2+1/3}{6}}{6}$ = $\frac{6^3-1}{6^3}$

We can keep doing this forever so the sum of any even number of reciprocals of natural numbers can represent $\frac{6^n-1}{6^n}$
Adding a last number of 1/6^n produces 1, so the sum of any odd number of reciprocals of 1atural numbers can be 1