Expected number of steps for reaching $K$ in a random walk

Question

Assuming steps are $+1/-1$ with a $50/50$ probability. What is the expected step count for reaching $10, 100$ or $K$?

Answer 1

Let $x$ be the expected number of steps to go from $0$ to $1$. Then, half the time you'll get there in one step, and the other half of the time you'll take one step and then need to cover twice the distance (because now you need to go from $-1$ to $1$). Therefore

$$ x = \frac{1}{2}(1) + \frac{1}{2}(1 + 2x) $$

which simplifies to

$$ x = x + 1 $$

which means that

$$ x = \infty $$

So if going from $0$ to $1$ in expectancy takes infinity steps, then going from $0$ to any other number also takes infinity steps.