Question

Use mathematical induction to prove the formula for all integers

n ≥ 1.
Let Sn be the equation 1 + 2 + 2^2 + 2^3 +... + 2^(n − 1) = (2^n) − 1
We will show that Sn is true for every integer n ≥ 1.
1 = (2^1 )− 1
The selected statement is true because both sides of the equation equal 1.
Show that for each integer k ≥ 1, if Sk is true, then Sk + 1 is true.
Assuming Sk is true, we have the following. (Simplify your answers completely.)
Sk = 1 + 2 + 22 + 23 +... +2^(k-1)=(2^k)-1
Then we have the following. (Simplify your answers completely.)
S_k + 1 = 1 + 2 + 22 + 23 +...+ 2k − 1 +__
= s_k+__
=(2^k)-

Answer 1

`S_(k+1) = 1 + 2 + 2^2 + 2^3 + ... + 2^(k-1)+2^k\\\\S_(k+1) = S_k+2^k\\\\S_(k+1) = 2^(k)-1+2^k\\\\S_(k+1) = 2^(k+1)-1`

This will mean:

• 1st empty box =
  `2^k`
• 2nd empty box =
  `2^k`
• 3rd empty box =
  `2^k`
• 4th empty box = 1