If x+\frac{1}{x}=a, then what is x^{n}+\frac{1}{x^{n}} equal to?

Guy Rollins

Guy Rollins

Answered question

2022-03-01

If x+1x=a, then what is xn+1xn equal to?

Answer & Explanation

chezpepina87j

chezpepina87j

Beginner2022-03-02Added 6 answers

Assuming x is a complex number, let x=eiϕ for some complex ϕ
Euler’s identity then tells us that x=cosϕ+isinϕ
Similarly, 1x=eiϕ, which is equal to cosϕisinϕ
Now, x+1x=(cosϕ+isinϕ)+(cosϕisinϕ)=2cosϕ, which is equal to a
This means cosϕ=a2
Or, ϕ=cos1a2
Again, xn can be written as eϕ=cosnϕ+isinnϕ and 1xn as eϕ=cosnϕisinnϕ
This means xn+1xn=2cosnϕ, which is equal to 2cos(ncos1a2)
And there we have it:
xn+1xn=2cos(ncos1a2)
benehmenshgf

benehmenshgf

Beginner2022-03-03Added 7 answers

If x+1x=a
then (x+1x)2=x2+2+1x2=a2
So x2+1x2=a22
Now (x2+1x2)(x+1x)=x3+x+1x+1x3=(a22)(a)
So x3+1x3=a33a
For n=4 we have
(a43a2)=(x3+1x3)(x+1x)=x4+x2+1x2+1x4
So we have x4+1x4=(a44a2+2)=(a22)22
Notice that, by first writing f(n)=xn+1xn, we actually have this pattern
f(n)=f(n1)af(n2)
Now lets find the form several more times:
f(5)=(a44a2+2)a(a33a)=a55a3+5a
f(6)=(a55a3+5a)a(a44a2+2)=a66a4+9a22

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