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Compute sum of fourth powers given the sum of lower powers

The sum of three numbers is 4, the sum of their squares is 10 and the sum of their cubes is 22. What is the sum of their fourth powers?

Solution Preview

Let's denote the three numbers by a1, a2, and a3. Consider the third degree polynomial:

p(x) = (1 + a1 x)(1 + a2 x)(1 + a3 x)

Take the logarithm:

Log[p(x)] =

Log(1 + a1 x) + Log(1 + a2 x) + Log(1 + a3 x)

Expand in powers of x by using that:

Log(1 + x) = x ...

Solution Summary

We show how one can efficiently compute the sum of the n-th powers (for integer n) of numbers if the sum of all k-th powers for 0< k < n (for integer k) of the numbers are given. We focus on the case where the sum of three numbers, their squares, and cubes are given but generalizing this to the general case is straightforward.