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Algebra Difficulty 5.1 AIME, harder Find the answer

35. Find the general term formula (involving complex numbers or trigonometric functions) for the following sequences:
 (1) 1,−1,1,−1,⋯ , \text { (1) } 1,-1,1,-1, \cdots,
(2) 1,0,1,0,⋯1,0,1,0, \cdots,
(3) 1,i,−1,−i,⋯1, i,-1,-i, \cdots,
(4) 1,0,0,1,0,0,1,⋯1,0,0,1,0,0,1, \cdots
(5) 1,0,0,0,1,0,0,0,1⋯1,0,0,0,1,0,0,0,1 \cdots.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

(Ans: (−1)n−1=cos⁡(n−1)π(-1)^{n-1}=\cos (n-1) \pi )
(2) 1+(−1)n−12=sin⁡2nπ2)\left.\frac{1+(-1)^{n-1}}{2}=\sin ^{2} \frac{n \pi}{2}\right)
(Ans: in−1i^{\mathrm{n}-1} )
(Ans 1+ωn−1+ω2(n−1)3\frac{1+\omega^{n-1}+\omega^{2(n-1)}}{3},
ω=−1+3i2)( Ans 1+in−1+(−1)n−1+(−i)n−14) \begin{array}{c} \left.\omega=\frac{-1+\sqrt{3} i}{2}\right) \\ \left(\text { Ans } \frac{1+i^{n-1}+(-1)^{n-1}+(-i)^{n-1}}{4}\right) \end{array}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.