Possible Duplicates:
Why is sum of a sequence $\displaystyle s_n = \frac{n}{2}(a_1+a_n)$?
Sum of n consecutive numbers
I know how to prove $\sum\limits_{i=1}^{n} i = \frac{n(n+1)}{2}$ works by mathematical induction, but how was the algorithm created?
Was it just trial and error?
How is any generic equation like this created with summation? At what level of math would I actually start learning how to come up with these equations myself?
Answer
A standard way of "discovering" this formula is to take the sum twice, in opposite order:
$$\begin{array}{ccccccccc}
1 & + & 2 & + & 3 & + & \cdots & + & n\\
n & + & n-1 & + & n-2 & + & \cdots & + & 1\\
\hline
(n+1) & + & (n+1) & + & (n+1) & + & \cdots & + & (n+1)
\end{array}$$
which readily yields that twice the sum equals $(n+1)n$, from which the result follows.
This kind of insight might strike "as a bolt from the blue" easily enough.
More interesting, perhaps, is to ask about more general formulas for sums of powers,
$$\sum_{i=1}^n i^k$$
with $k$ a positive integer. There are many systematic ways for finding such formulas; see for example this paper, or this Monthly paper by Beardon to get started.
No comments:
Post a Comment