1. ## Natural number

Find a where 2(1/a+1/(a+1)) is a natural number.

2. ## Re: Natural number

Originally Posted by louis33
Find a where 2(1/a+1/(a+1)) is a natural number.
$a=1 \implies 2\left(\dfrac{1}{a} + \dfrac{1}{a+1} \right) = 3$

3. ## Re: Natural number

Yeah, but I'd like to know how you got to that result in the first place. You can't just give me the answer

4. ## Re: Natural number

I just tried $a=1$ and it worked ...

5. ## Re: Natural number

$m=2\left(\frac{1}{a}+\frac{1}{a+1}\right)$

if $m$ is an integer then $m a-4=\frac{-2}{a+1}$ must be an integer

6. ## Re: Natural number

How did you get to that equation?

7. ## Re: Natural number

Originally Posted by louis33
Find a where 2(1/a+1/(a+1)) is a natural number.
It should be obvious that double the sum of two small fractions will be less than 1. In fact

$1 = 2 \left ( \dfrac{1}{4} + \dfrac{1}{4} \right ) \implies 1 > 2 \left ( \dfrac{1}{4 + k} + \dfrac{1}{5 + k} \right ) \text { if } k \ge 0.$

So you have exactly 3 numbers to test, 1, 2, and 3. And 2 and 3 don't work.