Math Is Fun Forum

simplyjasper

Member

Registered: 2009-11-15

Posts: 24

Fibonacci's Sequence

How do I prove by limits that this sequence is less than 2 for all values of n?

Last edited by simplyjasper (2009-11-16 06:20:03)

betterthangauss

Member

Registered: 2009-10-31

Posts: 11

Re: Fibonacci's Sequence

Use mathematical induction

since

Assume

Then

so

since

shows it cannot be negative

Hence, by induction we have

for all positive integers n

Last edited by betterthangauss (2009-11-16 06:55:51)

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Fibonacci's Sequence

Here's my solution, but I don't know if it involves limits in the way you were thinking

Last edited by Identity (2009-11-16 06:55:12)

rzaidan

Member

Registered: 2009-08-13

Posts: 59

Re: Fibonacci's Sequence

Hi simplyjasper,
You can prove that the sequence is less than 2 by mathematical induction together with finding the limit.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Fibonacci's Sequence

Hi simplyjasper;

Here is a proof by contradiction:

Suppose for some m that a(m) >= 2 then a(m+1)>=2:

If we square both sides of the recurrence we get:

a(m+1)^2 = a(m) +2

We run the recursion backwards to get:

a(m) = a(m+1)^2 - 2

Since a(m+1) >=2 then a(m) >=2.

We can now continue all the way back to a(1) >=2.

But this is a contradiction so a(n+1) = √(a(n) +2) is never >= 2.

Last edited by bobbym (2009-11-16 23:03:44)

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