Question

Let U1 =1, U2=1 and Un+2=Un+1+Un for n>=1.Use Mathematical Induction to show that:
Un=1/root 5 {(1+root5/2)n-(1-root5/2)n} for all n>=1. ​

Answer 1

Answer:

If F(n) is the Fibonacci Sequence, defined in the following way:

F(0)=0F(1)=1F(n)=F(n−1)+F(n−2)

I need to prove the following by induction:

F(n)≤(1+5–√2)n−1∀n≥0

I know how to prove the base cases and I know that the inductive hypothesis is "assume F(n)≤(1+5√2)n−1∀n≤k,k≥0".

For the inductive step, I need to show:

F(k+1)≤(1+5–√2)k+1

F(k)+F(k−1)≤(1+5√2)k -- by the definition of the Fibonacci sequence.

(1+5√2)k−1+(1+5√2)k−2≤(1+5√2)k -- by the inductive hypothesis?

If I could show this last line then I could prove the overall question but I am unsure of how to prove this and that last line may not be correct. Any help would be greatly appreciated!

Answer 2

Answer:

If F(n) is the Fibonacci Sequence, defined in the following way:

F(0)=0F(1)=1F(n)=F(n−1)+F(n−2)

I need to prove the following by induction:

F(n)≤(1+5–√2)n−1∀n≥0

I know how to prove the base cases and I know that the inductive hypothesis is "assume F(n)≤(1+5√2)n−1∀n≤k,k≥0".

For the inductive step, I need to show:

F(k+1)≤(1+5–√2)k+1

F(k)+F(k−1)≤(1+5√2)k -- by the definition of the Fibonacci sequence.

(1+5√2)k−1+(1+5√2)k−2≤(1+5√2)k -- by the inductive hypothesis?

If I could show this last line then I could prove the overall question but I am unsure of how to prove this and that last line may not be correct. Any help would be greatly appreciated!