Use the Principle of Mathematical Induction to prove that

Question

Use the Principle of Mathematical Induction to prove that 1 + 2^n≤ 3 ^n ∀𝑛 ≥ 1

Answer 1

Proving 3^(n+1)-2^(n+1)-1>0 is sufficient

The induction step goes

Something like this:

3^(n+1)-2^(n+1)-1

' adds and subtracts 2^n

>= 3^(n+1) -2^(n+1)+2^n-2^n-1

' 3^(n+1)>3^n

>=[3^n-2^n-1]+2^(n+1)+2^n

'By induction hypothesis, 3^n-2^n-2-1>0

>=0+2^(n+1)+2^n

'Factors

>= 2^n(2+1)

'Everthing is positive

= 2^n*3>0