2 + 4 + 8 + 16 + ... + 2ⁿ= 2ⁿ⁺¹ - 2

i found the answer here

• Letn= 1.Then:
• 2 + 22+ 23+ 24+ ... + 2n= 21= 2
• ...and:
• 2n+1- 2 = 21+1- 2 = 22- 2 = 4 - 2 = 2
• So (*) works forn= 1.
• Assume, forn=k, that (*) holds; that is, that
• 2 + 22+ 23+ 24+ ... + 2k= 2k+1- 2
• Letn=k+ 1.
• 2 + 22+ 23+ 24+ ... + 2k+ 2k+1
• = [2 + 22+ 23+ 24+ ... + 2k] + 2k+1
• = [2k+1- 2] + 2k+1
• = 22k+1- 2
• = 212k+1- 2
• = 2k+1+1- 2
• = 2(k+1)+1- 2
• Then(*) worksforn=k+ 1.

im lost in the part of

= [2 + 22+ 23+ 24+ ... + 2^k] + 2^k+1

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

= 22^k+1- 2

= 2¹2^k+1- 2

= 2^k+1+1- 2

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

can someone please explain what happen on each line?