If i^2=-1, then the sum i+i^2+i^3+... upto 1000 terms is equal to… - Vidyadip

MCQ

If $\text{i}^2=-1,$ then the sum $\text{i}+\text{i}^2+\text{i}^3+...$ upto $1000$ terms is equal to:

A
$1$
B
$-1$
C
$i$
✓
$0$

✓

Answer

Correct option: D.

$0$

$\text{i}+\text{i}^2+\text{i}^3+\text{i}^4...\text{i}^{1000}$
$\text{i}+\text{i}^2+\text{i}^3+\text{i}^4$
$[\because\text{i}^2=-1,\text{i}^3=-\text{i}$ and $\text{i}^4=1]$
$=\text{i}-1-\text{i}+1$
$=0$
Similarly, the sum of the next four terms of the series will be equal to $0.$
This is because the powers of $i$ follow a cyclicity of $4$.
Hence, the sum of all terms, till $1000$, will be zero.
$\text{i}+\text{i}^2+\text{i}^3+\text{i}^4...\text{i}^{1000}=0$

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