How is this true? Mathematical induction proof

Question

so I have $1/2 + 1/4 + 1/8 + \ldots + 1/2^n = 1 - 2^{-n}$ to prove by mathematical induction. I did all the steps and I end up with:

$$1-2^{-n}+2^{-n-1}=1-2^{-n-1}$$

I tried this out in Wolfram and another "calculator" and it says that it's true. How? How do I transform this so it's equal?

---

EDIT: Done and understood. Thank you all!

Answer 1

$$1-2^{-n}+2^{-n-1}=1-2\cdot2^{-n-1}+2^{-n-1}=1-2^{-n-1}$$

Answer 2

If we have the difference of two consecutive powers of two, the smaller number is half the larger, so the answer is the smaller one, for example $16 - 8$ is just $8$ but it works for negative exponents too.