Simple Induction Interesting Algebra Problem

PeroK
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I found this interesting video from Presh Talwalkar:

Problem Statement. If:
$$x + y + z = 1$$$$x^2 + y^2 + z^2 = 2$$$$x^3 + y^3 + z^3 = 3$$ Then, find the value of the higher powers such as $$x^5 + y^5 + z^5$$
The solution posted there uses the full Girard-Newton Identities. Here is an elementary solution using the same ideas:
First, note that $$x^2 + y^2 + z^2 = (x + y + z)^2 - 2(xy + xz + yz)$$ and, for ##n \ge 3## (which is easy to verify): $$x^n + y^n + z^n = (x + y +z)(x^{n-1} + y^{n-1} + z^{n-1}) - (xy + xz + yz)(x^{n-2} + y^{n-2} + z^{n-2}) + xyz(x^{n-3} + y^{n-3} + z^{n-3})$$ Plugging the values we have in the first equation gives:
$$2 = 1 - 2(xy + xz + yz)$$ Hence $$xy + xz + yz = -\frac 1 2$$ Then, for ##n = 3## we have:
$$3 = (1)(2) - (-\frac 1 2)(1) + xyz(3)$$ Giving $$xyz = \frac 1 6$$ Then, for ##n = 4## we have:
$$x^4 + y^4 + z^4 = 3 - (-\frac 1 2)(2) + \frac 1 6 = \frac{25}{6}$$ And, for ##n = 5## we have:
$$x^5 + y^5 + z^5 = \frac{25}{6} - (-\frac 1 2)(3) + (\frac 1 6)(2) = 6$$
In general we have $$x^n + y^n + z^n = (x^{n-1} + y^{n-1} + z^{n-1}) + \frac 1 2(x^{n-2} + y^{n-2} + z^{n-2}) + \frac 1 6 (x^{n-3} + y^{n-3} + z^{n-3})$$ And that, in fact, ##x^n + y^n + z^n## must be rational for all ##n##.

Finally, note that we can generalise this procedure for any initial values of the first three expressions.
 
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WolframAlpha only finds a complex solution. All the steps are possible with complex numbers, of course.
x = 1.431, y = -0.215-0.265i, z = -0.215+0.265i
 
mfb said:
WolframAlpha only finds a complex solution. All the steps are possible with complex numbers, of course.
x = 1.431, y = -0.215-0.265i, z = -0.215+0.265i
Yes, I forgot to add a note to say that the underlying solutions for ##x, y, z## are complex. Early in the video he uses Wolfram Alpha as well!
 
I believe there is a significant gap in the availability of resources that emphasize the underlying logic of abstract mathematical concepts. While tools such as Desmos and GeoGebra are valuable for graphical visualization, they often fall short in fostering a deeper, intuitive understanding. Visualisation, in this sense, should go beyond plotting functions and instead aim to reveal the reasoning and common-sense foundations of the concept. For example, on YouTube one can find an excellent...

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