prove by induction

Question

Prove by Prove, using induction that

2^(n) ≤ 2^(n+1) − 2^(n−1) − 1

 

 

for all n ≥ 1

 

Answer 1

2ⁿ ≤ 2ⁿ⁺¹ − 2^n-1 − 1

 

Is it true for n = 1?

 

2¹ ≤ 2² - 2⁰ - 1

2 ≤ 4 - 1 - 1

2 ≤ 2

 

is it true for n?

2ⁿ ≤ 2(2ⁿ) - (1/n)2ⁿ - 1

0 ≤ 2ⁿ - (1/2)2ⁿ - 1

1 ≤ (1/2)2ⁿ

2 ≤ 2ⁿ   Which is true for all n ≥ 1

 

If true for n (Yes), is it true for n+1

2ⁿ⁺¹ ≤ 2ⁿ⁺² - 2ⁿ - 1

2¹ ≤ 2² - 1 - 1/2ⁿ

2 + 1 + 1/2ⁿ ≤ 2²

3 + 1/2ⁿ ≤ 4

1/2ⁿ ≤ 1

which is true for every n ≥ 0 , and n ≥ 1 anyway

 

having established

1) true for 1

2) if true for n, then true for n+1

 

we have met the requirements for proof by mathematical induction.