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Solution

(related to Problem: Sums of Falling Factorial Powers)

Using the formula for the falling factorial powers from antidifferences of some functions, we get

$$\sum_{x=3}^{16}x^{\underline{3}}=\frac{x^{\underline 4}}{4}\;\Rule{1px}{4ex}{2ex}^{17}_{3}=\frac{17\cdot 16\cdot 15\cdot 14-3\cdot 2\cdot 1\cdot 0}{4}=14280.$$

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References

Bibliography

1. Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960