An algebra problem by Priyanshu Mishra
If x + 1/x = -1 Find the value of x^99 + 1/x^99.
The answer is 2.
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2 solutions
I just noted that when x = e i y , cos y = − 2 1 which easily gives y = 3 2 π . From then you can just conclude x 9 9 + x 9 9 1 = 2 cos 9 9 y = 2 which seems heaps easier.
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oh yea that is quicker. I suppose the fact that Euler's Formula can be used to prove De Moivre's Theorem, means that they both work.
Nice solution Curtis. Have you ever appeared for International Mathematical Olympiad (IMO) , conducted by Mathematical Association Of America ( MAA) ?
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I assumed x as a complex cube root of unity since it satisfies the condition of x^2+x=-1. We know that w^3=1(where w is the complex cube root of unity).hence w^99=1. So the expression reduces to 1+1=2(the required ans)
from x + 1/x = -1, x^2+x+1=0.
x=-1^2/3.
thus the required answer is -1^66+1/-1^66=1+1=2.
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x = ( − 1 ) 2 / 3 , is not the solution to x 2 + x +1 = 0, so your argument is incomplete/wrong (even if it does reach the correct answer). Take a look at my solution to see how you get x 9 9 + x 9 9 1 = 2
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Thanks for your solution. It helped me a lot.
Multiply the given equation by x and rearrange to give: x 2 + x +1 = 0. Using the quadratic equation gives x = - 2 1 ± 2 3 j , where j = − 1 . Now I will apply De Moivre's Theorem, for the positive conjugate solution: x 9 9 =cos[99. 3 2 π ] + j sin[99. 3 2 π ] = cos66 π + j sin66 π = 1, as 66 π ≡ 0 mod{2 π }. Hence, the answer is 2 . (Note: the same method can be applied to the negative conjugate to produce the exact same answer)