How to find the closed form definition of a series?

Question

How does one determine if the following series is arithmetic or geometric? 

 

n  f(n)  Σ
0  1     1
1  5     6
2  14   20 
3  30   50
4  55   105

 

If it is arithmetic, what is the common difference? If it is geometric, what is the common ratio? Using this information, I have to find the closed-form definition of the series, but I can't seem to find a common difference or a common ratio...

Answer 1

 

You are on the right track; just keep thinking:

 

n  f(n)  Σ          difference           ratio
0  1     1
1  5     6             5-1=4              5/1=5
2  14   20           14-5 = 9           14/5=2.8
3  30   50           30-14 = 16        30/16 = 1.875
4  55   105         55-39 = 25        55/30 = 1.833

 

Since there is not a common difference between terms, it is not an Arithmetic Sequence.

Since there is not a common ratio between terms, it is not a Geometric Sequence.

 

There does, however, seem to be a pattern when we compute the differences between terms

   [looks like the difference between term (n) and term (n-1) is the square of [n+1)],

   so,   a_n = (n+1)2 + a_n-1      for n>0

          a_n= 1       for n=0

     [note: this is the recursive definition of the series.]

 

Each f(n) is the sum of the squares of the numbers from 1 (or from 0) to (n+1).  For example, f(4) = 1² + 2² + 3² + 4² +5²= 1 + 4 + 9 + 16 + 25 = 55.

 

A formula for the sum of squares can be derived in different ways (depending on your level of math).  See:

http://www.trans4mind.com/personal_development/mathematics/series/sumNaturalSquares.htm

 

It is written as:      f(n) = (n/6)(n+1)(2n+1).

 

Be sure to replace this n with the (n+1) in this problem (for example our f(4) is the sum of squares of numbers 0 to 5):

                f(n) = ((n+1)/6)(n+2)(2n+3)

       and, for example,   f(4) = (5/6)(6)(11) = 55

Answer 2

If you look at f(n) - f(n-1) for each n you should see a pattern.