Prove that 22n−1 is divisible by 3, ∀ integers n⩾1
Show p(a) [’a’ is min value of statement]
3|22(1)−1
22−1=3(i) [i is an integer]
3 = 3(i)
Suppose p(k) k = n
22k−1=3(i) [inductive step]
We wish to show p(k + 1)
22(k+1)−1=3(i)
[starting on left hand side]
22(k+1)−1=22k+2−1
= 2222k−1
= 22k∙4−1
= ???????
How do I get there from here, please provide your reasoning. Thanks!
Show p(a) [’a’ is min value of statement]
3|22(1)−1
22−1=3(i) [i is an integer]
3 = 3(i)
Suppose p(k) k = n
22k−1=3(i) [inductive step]
We wish to show p(k + 1)
22(k+1)−1=3(i)
[starting on left hand side]
22(k+1)−1=22k+2−1
= 2222k−1
= 22k∙4−1
= ???????
How do I get there from here, please provide your reasoning. Thanks!