recursive function

Question

if f(3)=5 and f(4)=8

 

Answer 1

In an Arithmetic Sequence, each term is separated from the previous term by a "common difference," called d.

 

If f(3)=5 and f(4)=8, then d=3.

 

The recursive definition of a sequence gives the formula for each term, f(n), as a function of the previous term or other previous terms -- that is called the recursive case.  The base case is the term that has a formula which does not refer to other terms in the sequence.

 

For the sequence,   -1, 2, 5, 8, 11, 14, ...     [note: f(3)=5 and f(4)=8 and d=3; start with f(1)]

   the recursive definition is:

         f(1) = -1                         [the base case]

         f(n) = f(n-1) + 3             [the recursive case]

 

Here is how you would use this recursive definition to evaluate f(4):

      f(4) = f(3) + 3

      f(4) =  (f(2)+3) + 3         [because f(3)=f(2)+3]

      f(4) = ((f(1)+3)+3 + 3)

      f(4) = ((-1+3)+3 + 3)      [use base case:  f(1)=-1]

      f(4) = ((2)+3+3)

      f(4) = 8                         [now, it is easy to calculate]

 

For an exercise, PLZ calculate f(6) and see if you get 14.

Answer 2

It's the fibonnaci sequence

 

1,1,2,3,5,8,13,21,34,Etc

 

The next term in the sequence is the sum of the previous two terms