What is the nth term for the Fibonacci sequance?

Answer 1

The Fibonacci sequence is generally defined recursively, as described below. The recursive form is easy to use but is a long process for large n. On the other hand, while there is a formula for the nth term, it is not easy to understand or use!

The recursive form is as follows:

f(1) = 1

f(2) = 1

f(n+2) = f(n) + f(n+1) for n = 1, 2, 3, ...

To find the nth term, you need to know the combinatorial formula, nCr [should be superscript n and subscript r, but try doing that in this rubbish browser!]

nCr = n!/[r!*(n-r)!]

By the way, n! = 1*2*3* ...*n and also, 0! = 1 (by definition).

Then, if k is the integer part of (n-1)/2,

f(n) = sum, from r = 0 to r = k of (n-r-1)Cr

For example, if n = 6 then k = int[(n-1)/2] = int(5/2) = 2.

So f(6) = (6-0-1)C0 + (6-1-1)C1 + (6-2-1)C2 [the summation is up to r = k = 2.]

= 5C0 + 4C1 + 3C2

= 1 + 4 + 3 = 8.