How can I prove this mathmatical induction?

Question

∑_j=1ⁿ 2^j = 2(2ⁿ-1)

Answer 1

Let n=1 then the sum = 2¹ = 2 2(2ⁿ-1) also = 2 for n=1

Let n=2 then the sum adds 2² to 2 to get 6 2(2²-1) also = 6

Let n=3 then the sum adds 2³ to 6 to get 14 2(2³-1) also = 14

That's the basic step, a little more than necessary, but it shows what's going on better

Now assume the sum of n terms of 2ⁿ = 2(2ⁿ-1) = 2ⁿ⁺¹-2,

Assume it works for n terms, then if so, show it works for n+1 terms

Show the sum of n+1 terms of 2ⁿ⁺¹=2ⁿ⁺²-2 or 2(2ⁿ⁺¹-1)

we assumed 2+2²+2³...+2ⁿ = 2ⁿ⁺¹-2

add 2ⁿ⁺¹ to both sides

2+2²+....2ⁿ⁺¹ = 2ⁿ⁺¹ -2 + 2ⁿ⁺¹ = 2(2ⁿ⁺¹) - 2 = 2ⁿ⁺²-2 = 2(2ⁿ⁺¹-1)

QED

If it's true for n=1 and n=2, and it's true for any n terms, then it must be true for n+1 terms.

We know it works for n=2, so it must be true for n=3, and since it's true for n=3 then it's also true for n=4

and that process has no end. It's true for any integer n.