My Contribution to the Mathematical Arts
Many, many years ago, sometime in high school, I learned that the formula for computing
![Figure1](figure1.png)
is
![Figure2](figure2.png)
I'm pretty sure we were shown the geometrical derivation of this formula. In college, in late 1975 or early '76, in a Math Lab one problem asked to guess the formula for
![Figure3](figure3.png)
In my handwritten lab notebook I used induction to show that the solution to (3) is:
![Figure4](figure4.png)
Having solved this specific case, I wanted to see if there was a general solution for any positive integer n and any positive integer exponent. Based on equations (2) and (4), I conjectured that the general solution would be a polynomial of this form:
![Figure5](figure5.png)
The derivation used induction on the general formula and found that the coefficients to the solution are:
![Figure6](figure6.png)
Computed coefficients for 0 <= k <= 10 are here.
Perhaps of interest are these properties of the coefficients:
![Figure7](figure7.png)
![Figure8](figure8.png)
![Figure9](figure9.png)
![Figure10](figure10.png)
![Figure11](figure11.png)
![Figure1](figure1.png)
is
![Figure2](figure2.png)
I'm pretty sure we were shown the geometrical derivation of this formula. In college, in late 1975 or early '76, in a Math Lab one problem asked to guess the formula for
![Figure3](figure3.png)
In my handwritten lab notebook I used induction to show that the solution to (3) is:
![Figure4](figure4.png)
Having solved this specific case, I wanted to see if there was a general solution for any positive integer n and any positive integer exponent. Based on equations (2) and (4), I conjectured that the general solution would be a polynomial of this form:
![Figure5](figure5.png)
The derivation used induction on the general formula and found that the coefficients to the solution are:
![Figure6](figure6.png)
Computed coefficients for 0 <= k <= 10 are here.
Perhaps of interest are these properties of the coefficients:
![Figure7](figure7.png)
![Figure8](figure8.png)
![Figure9](figure9.png)
![Figure10](figure10.png)
![Figure11](figure11.png)
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