can anyone tell me if this seems correct? been working on this for a while now. Just want to make sure. I struggle with these quite a bit.

1.) 2 + 2^2 + 2^3 + 2^4 + ... + 2^n = 2^n+1 -2

Let n=1

2 + 2^2 + 2^3 + 2^4 + ... + 2^n = 2^1 = 2

and

2^n+1 -2 = 2^1+1 - 2 = 2^2 -2 = 4 - 2 = 2

assume n=k

2 + 2^2 + 2^3 + 2^4 + ... + 2^k = 2^k+1 -2

Let n=k+1

2 + 2^2 + 2^3 + 2^4 + ... + 2^k = 2^k+1 -2

= [2 + 2^2 + 2^3 + 2^4 + ... + 2^k] + 2^k+1

= [2^k+1 -2] + 2^k+1

= 2 x 2^k+1 - 2

= 2^1 x 2^k+1 - 2

= 2^k+1+1 - 2

= 2^(k+1)+1 - 2