Lovely product

Algebra Level 3

For i = 1 i = \sqrt{-1} , evaluate n = 1 100 i n \large \displaystyle \prod_{n=1}^{100} i^n .

1 1 i i 1 -1 i -i

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2 solutions

Aareyan Manzoor
Dec 18, 2015

this is weird solution, but shared any ways. note that i and its powers are roots of x 4 1 = 0 x^4-1=0 . by vietas the product of four taken at a time is -1. we are multiplying minus one 25 times, so the answer is -1.

Rohit Udaiwal
Dec 18, 2015

ι 1 ι 2 ι 3 ι 100 = ι 1 + 2 + 3 + 100 = ι 100 × 101 2 = ι 5050 = ι 2 = 1. \iota^1 \cdot \iota^2 \cdot \iota^3 \ldots \iota^{100} \\=\iota^{1+2+3\ldots+100}=\iota^{\frac {100×101}{2}}=\iota^{5050}\\ =\iota^2=-1.

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