4.(a) Prove that
(−1 + i√3)^n + (−1 − i√3)^n
has either the value 2^n+1 or the value −2^n. if n is any integer (positive, negative or zero).
(b) The complex numbers z1 and z2 are connected by the relation
z1 = z2 +1/z2
If the point representing z2 in the Argand diagram describes a circle of radius a and centre at the origin, show that the point representing z1 describes the ellipse
x^2/(1 + a^2)^2 + y^2/(1 − a^2)^2=1/a^2.
1
Expert's answer
2020-07-01T18:50:47-0400
ANSWER
(a) Let z=(−1+i3)∣z∣=1+3=2,φ=argz=32π , z=−1−i3 .
Therefore : z=2(cos32π+isin32π),z=2(cos32π−isin32π) . By the Moivre's formula, we have (−1+i3)n+(−1−i3)n=2n(cos32nπ+isin32nπ+cos32nπ−isin32nπ)==2n+1cos32nπ=⎩⎨⎧−2n,ifn=3k−2−2n,ifn=3k−12n+1,ifn=3k , because cos32(3k−2)π=cos(2kπ−34π)=cos(π+3π)=−cos3π=−21 , cos32(3k−1)π=cos(2kπ−32π)=cos(π−3π)=−cos3π=−21 .
So, (−1+i3)n+(−1−i3)n =2n+1 or −2n if n is any integer.
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