Question

the numbers n, m and k are natural numbers. Express each of the following as a natural number. 3^n+3/3^n+1

Answer 1

To express the given expression as a natural number, let's simplify it step by step.

The expression is:

`\displaystyle\sf (3^n+3)/(3^(n+1))`

We can rewrite the numerator as:

`\displaystyle\sf 3^n + 3 = 3^n + 3 * 1 = 3^n + 3 * 3^0`

Now, let's combine the terms in the numerator:

`\displaystyle\sf 3^n + 3 * 3^0 = 3^n + 3 * 3^0 = 3^n + 3`

Next, let's rewrite the denominator as:

`\displaystyle\sf 3^(n+1) = 3^n * 3^1 = 3^n * 3`

Now, substituting the rewritten numerator and denominator back into the expression, we have:

`\displaystyle\sf (3^n + 3)/(3^(n+1)) = (3^n + 3)/(3^n * 3) = (3^n)/(3^n) + (3)/(3^n)`

Simplifying further, we get:

`\displaystyle\sf (3^n)/(3^n) + (3)/(3^n) = 1 + (3)/(3^n)`

So, the expression
`\displaystyle\sf (3^n+3)/(3^(n+1))` simplifies to
`\displaystyle\sf 1 + (3)/(3^n)`, which can be expressed as a natural number.