First slide

Algebra of complex numbers

Question

The sequence $S = i + 2 i^{2} + 3 i^{3} + 4 i^{4} + \dots$ up to 100 terms simplifies to, where $i = \sqrt{- 1}$

Moderate

A

$50 (1 - i)$
B

$25 i$
C

$25 (1 + i)$
D

$100 (1 - i)$

Solution

$S = i + 2 i^{2} + 3 i^{3} + \dots + 100 i^{100}$
$S \cdot i = i^{2} + 2 i^{3} + \dots + 99 i^{100} + 100 i^{101}$
$\begin{array}{l} S (1 - i) = i + i^{2} + i^{3} + \dots + i^{100} - 100 i^{101} \\ S (1 - i) = (i + i^{2} + i^{3} + i^{4}) + (i^{5} + i^{6} + i^{7} + i^{8}) \\ \Rightarrow S (1 - i) = 0 + 0 + \dots + (i^{97} + i^{98} + i^{99} + i^{100}) - 100 i \\ \Rightarrow S = \frac{- 100 i}{1 - i} = \frac{- 100 i (1 + i)}{(1 - i) (1 + i)} \\ \Rightarrow S = - 50 (i - 1) \\ ∴ S = 50 (1 - i) \end{array}$

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