Complex Numbers and Quadratic Equations — MATHS STD 11 Science — Question

4. 0

Solution:

$\text{i}+\text{i}^2+\text{i}^3+\text{i}^4...\text{i}^{1000}$

$\text{i}+\text{i}^2+\text{i}^3+\text{i}^4 \ [\because\text{i}^2=-1,\text{i}^3=-\text{i} \ \text{and} \ \text{i}^4=1]$

$=\text{i}-1-\text{i}+1$

$=0$

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.

$\text{i}+\text{i}^2+\text{i}^3+\text{i}^4...\text{i}^{1000}=0$