Proof by induction of a sum.

Question

I am at the step where I am proving $P(k+1)$:

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

How am I going to make these equal?

Ps: Just realized this is just an exponent rule, I need coffee.

Answer 1

Notice: $$\sf \color{red}{2^k}+\color{red}{2^k}-1=2(2^k)-1=2^1(2^k)-1=2^{k+1}-1.$$ Where we used the neat property: $$\sf a^m\cdot a^n=a^{m+n}.$$

Answer 2

Notice looking at the left side of the "equality", we have $$ 2^k-1+2^k=2^k+2^k-1=2^k(1+1)-2=2^k\cdot 2 -1=2^{k+1}-1 $$ This gives us $$ 2^{k+1}-1=2^{k+1}-1 $$ Which is trivially true. So we see that the two sides of the equality are indeed the same.