determine a variable,under a given conditionIf a natural number is (2n+1)/n, how to calculate n, that is also natural guessing n is a solution?
If (2n+1)/n is a natural number, then 2n+1 is divisible by n.
Then we will rewrite as follow.
2n+1)/n = k where k is a natural number.
==> (2n/n + 1/n ) = k
==> 2 + 1/n = k
Now since k is a natural number, then both terms are natural numbers.
We know that 2 is a natural number.
Also, 1/n must be a natural number.
==> In order for 1/n to be a natural number, the only option is that n= 1
==> 2 + 1 = k = 3
==> (2n+1)/n = 3
"To guess" is not a mathematical way of solving.
If (2n+1)/n is a natural number, then (2n+1) is divided by n.
We'll write 2n + 1 = n + n + 1
(2n+1)/n = (n + n + 1)/n
(n + n + 1)/n = n/n + n/n + 1/n
(n + n + 1)/n = 1 + 1 + 1/n
(n + n + 1)/n = 2 + 1/n
For (2n+1)/n to be natural, the ratio 1/n has to be a natural number too.
1/n is natural if and only if n is a natural divisor of 1, so n = 1
(2n+1)/n = 2 + 1 = 3