Sum of non-geometric, non-arithmetic series

[sayeo87]

Homework Statement

How should I find the nth sum, of the series 2 + 5 + 9 + 14 + ... + n ?
(the difference between terms increases by 1 each time)

Homework Equations

The Attempt at a Solution

I only know how to sum geometric and arithmetic series and this is neither. Perhaps somebody could at least give me the name of this series so I can look it up on the net as a start? Thanks.

 

[tiny-tim]

How should I find the nth sum, of the series 2 + 5 + 9 + 14 + ... + n ?
(the difference between terms increases by 1 each time)

Hi sayeo87!

Hint: a_n - a_n-1 = n + 1

so a_n = a₁ + … ?

 

[statdad]

Do you want the $$n-\text{th}$$ sum (sum of the terms from $$1 \text{ to } n$$ or the $$n-\text{th}$$ term (formula to generate the next entry from one or more previous terms)?

 

[HallsofIvy]

Although you probably have not learned it yet, there is a general method for problems like this: Newton's divided difference method.

Set up a diagram like this:
$$\begin{array}{ccccc}n & f(n) & \Delta f & \Delta^2 f & \Delta^3 f \\0 & 0 & 2 & 3 & 1 \\ 1 & 2& 5 & 4 & 1 \\ 2 & 7 & 9 & 5 & 1 \\ 3 & 16 & 14 & 6 & _ \\ 4 & 30 & 20 & _ & _ \\ 5 & 50 & _ & _ & _\end{array}$$

That is, the first column is the non-negative integers, the second column, f(n), is the partial sum up to that n, the third column is the difference between two consecutive numbers in the second column (and is just the individual terms in the series), the fourth column is the difference between two consecutive numbers in the third column, and the fifth column is the difference between two consecutive numbers in the fourth column. We can stop there because all numbers in the fourth column are the same so all succeeding columns will be all zeros. I put in the n= 0 row by working backward from the last column.

Now, Newton's divided difference formula is an analog of Taylor's series for continuous functions, using "$\Delta$" instead of the derivative:
[tex]f(n)= f(0)+ \Delta f(0) n+ \frac{\Delta^2 f(0)}{2!}n(n-1)+ \frac{\Delta^3 f(0)}{3!}n(n-1)(n-2)[/itex]<br /> <br /> For this particular sequence we have <br /> f(n)= 0+ 2n+ (3/2)n(n-1)+ (1/6)n(n-1)(n-2)<br /> <br /> Multiply that out and you will get a cubic formula (or product of three linear terms) that will give the nth sum.[/tex]