How do I write $2(2^k+1) - 1$ as $(2 \cdot 2^k + 1)$?

Question

How do I write $2(2^k+1) - 1$ as $(2 \cdot 2^k + 1)$?

Mathematically, it is equivalent. But I need to the former form into the latter form for step 2 of inductive step for mathematical induction problem.

Answer 1

Distributing out the $2$ we obtain: $$2(2^k+1)-1 = (2^k \cdot 2 +2)-1 = 2^k \cdot 2 +2-1 = 2\cdot2^k+1$$

Answer 2

$$ 2\Big( 2^k + 1\Big) - 1 = \Big((2\cdot2^k) +(2\cdot1)\Big) - 1 = (2^{k+1} + 2) - 1 = 2^{k+1}+(2-1) = \cdots $$