Simplify $\frac{2}{i} + \frac{3}{i^{3}} + \frac{4}{i^{4}} + \frac{5}{i^{5}}$ in the form of $a + i b$ & represent graphically & find modulus & amplitude.

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Solution

Given $\frac{2}{i} + \frac{3}{i^{3}} + \frac{4}{i^{4}} + \frac{5}{i^{5}} = a + i b$
$\Rightarrow i^{2} = - 1$ $i^{3} = - - i$
$i^{4} = - 1$ $i^{5} = - i$
$\Rightarrow \frac{- 2 \times - 2}{i} + \frac{3}{(- i)} + 4 + \frac{5}{i} = a + i b$
$\Rightarrow - 2 i + 3 i + 4 - 5 i = a + i b$
$4 - 4 i = a + i b$
$\Rightarrow a = 4,$ $b = - 4$

$\Rightarrow | a + i b | = \sqrt{4^{2} + 4^{2}} = 4 \sqrt{2}$
$\Rightarrow a r g (a + i b) = - tan (\frac{4}{4})$ ( as it is 4th quadrant )
$\Rightarrow a r g (a + i b) = \frac{- π}{4}$
Hence, the answer is $\frac{- π}{4} .$

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