Adding always the half of the previous number to a number beginning with 1... This question blows my mind.

Let's say we start at 1 and keep adding always 0.5 * the previous number.

1 + 0.5 + 0.25 + 0.125 + ..... 

The question is: will it reach infinity?

I really don't know, kept thinking about this alot.
Rewrite it with fractions and it might be a bit clearer:

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} ...$

Now calculate the partial sums:

$1$, $1 \frac{1}{2}$, $1 \frac{3}{4}$, $1 \frac{7}{8}$, $1 \frac{15}{16}$, ...

See a pattern?  Can you see this going over $2$?  How close do you think it might get?
So what you're looking for is the value of this series, which will call it $S$.

$$S=\sum_{n=0}^{\infty} \left( \frac{1}{2} \right)^n=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$$

If we divide this value $S$ by $2$ we get

$$\frac{S}{2}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...=S-1 \Longrightarrow S=2S-1$$

so now with simple algebra we get the result that the sum of these numbers $S$ tends to $2$, which is not $\infty$.