Ends at 30

Algebra Level 3

i − i 2 − i 3 − i 4 − ⋯ − i 30 = ? \large i-i^2-i^3-i^4- \cdots -i^{30}= ?

Notation: i = − 1 i = \sqrt{-1} denotes the imaginary unit .

2 + i 2+i i i 1 + i 1+i 0 0 3 i 3i − 1 -1 1 1

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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1 solution

Chew-Seong Cheong
Oct 23, 2017

z = i − i 2 − i 3 − i 4 − ⋯ − i 30 = i − ∑ k = 2 30 i k = i − i 2 ∑ k = 0 28 i k = i − ( − 1 ) ( i 29 − 1 i − 1 ) = i + ( i − 1 i − 1 ) = i + 1 \begin{aligned} z & = i - i^2 - i^3 - i^4 - \cdots - i^{30} \\ & = i - \sum_{k=2}^{30} i^k \\ & = i - i^2 \sum_{k=0}^{28} i^k \\ & = i -(-1) \left(\frac {i^{29}-1}{i-1}\right) \\ & = i + \left(\frac {i-1}{i-1}\right) \\ & = \boxed{i+1} \end{aligned}

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