First slide

Geometric progression

Question

$i - 2 - 3 i + 4 \dots to 100 terms =$

Moderate

A

$50 (1 - i)$
B

$25 i$
C

$25 (1 + i)$
D

$100 (1 - i)$

Solution

$\begin{array}{l} S = i + 2 i^{2} + 3 i^{3} +, \dots, + 100 i^{100} \\ S \cdot i = i^{2} + 2 i^{3} +, \dots, + 99 i^{100} + 100 i^{101} \\ ∴ S (1 - i) = (i + i^{2} + i^{3} +, \dots, + 100 terms) - 100 i^{101} \\ = \frac{i (1 - i^{100})}{1 - i} - 100 i = \frac{i (1 - 1)}{1 - i} - 100 i = - 100 i \\ ∴ S = \frac{- 100 i}{1 - i} = \frac{- 100 i (1 + i)}{2} = 50 (1 - i) \end{array}$

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