Evaluate

Question:

Evaluate $\left(i^{4 n+1}-i^{4 n-1}\right)$

 

Solution:

We have, $i^{4 n+1}-i^{4 n-1}$

$=i^{4 n} \cdot i-i^{4 n} \cdot i^{-1}$

$=\left(i^{4}\right)^{n} \cdot i-\left(i^{4}\right)^{n} \cdot i^{-1}$

$=(1)^{n} \cdot i-(1)^{n} \cdot i^{-1}$

$=i-i^{-1}$

$=\mathrm{i}-\frac{1}{\mathrm{i}}$

$=\frac{\mathrm{i}^{2}-1}{\mathrm{i}}$

$=\frac{-1-1}{\mathrm{i}}$

$=\frac{-2}{\mathrm{i}} \times \frac{\mathrm{i}}{\mathrm{i}}$

$=\frac{-2 \mathrm{i}}{\mathrm{i}^{2}}=\frac{-2 \mathrm{i}}{-1}$

$=2 \mathrm{i}$

 

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