x is a square number. Show that the next square number is x + 2√x + 1.

This is an algebraic proof. Let `x` be a square number. By definition, this means that `x` is the result of squaring an integer. Let this integer be `n`. So, `x = n²`. From this, we can also say that the square root of `x` is `n`: `√x = √(n²) = n`. The next integer after `n` is `n + 1`. The next square number after `x` (which is `n²`) will be the square of the next integer, `(n + 1)`. So, the next square number is `(n + 1)²`. Let's expand this expression: `(n + 1)² = (n + 1)(n + 1) = n² + 2n + 1`. Now we need to show that this is the same as `x + 2√x + 1`. We can do this by substituting our definitions of `x` and `√x` back into our result: - Replace `n²` with `x`. - Replace `n` with `√x`. So, `n² + 2n + 1` becomes `x + 2√x + 1`. This completes the proof.