When I was asked to *prove* the following power summation formulas by mathematical induction:

,

,

,

I wondered how these closed forms are obtained in the first place! Did a little bird whisper the formulas into our ears?

Even though there are elementary derivations of the power summation formulas, for example, the visual derivations in *Proof Without Words by MAA, *none goes beyond the 3rd power.

In this blog, I will construct a recursive generator capable of generating the closed form of power summation

,

for *all* .

Let us start with a picture.

Fig. 1

Let denotes the area of a rectangle, Fig.1 shows

.

Since

,

,

,

we have

.

when , (1) becomes

,

hence,

,

or,

.

When p = 2,

.

Substituting (2) for , we have

or,

.

In general, ,

.

Solving for , we obtain

.

Let

,

then

What we have here is a recursive generator capable of generating power summation formulas for virtually all integrer powers. Implemented in Omega CAS Explorer, the figures below illustrate the awesomeness of this generator:

Fig. 2

Fig. 3