Question

please give the correct answer it's humble request ❤❤❤❤❤​

Answer 1

$\underline{\bold{\red{Question} }} :$

$\sf \: Prove \: that : \quad 1 + i ^{10} + i ^{20} + i ^{30} = 0$

$\underline{| \bold{\green{Answer} }} :$

$\begin{aligned} \mathbb{LHS} \\ \\ = \: & \sf1 + {i}^{10} + {i}^{20} + {i}^{30} \\ \\ = \: & \sf1 + {( {i}^{2} )}^{5} + {( {i}^{2} )}^{10} + {( {i}^{2} )}^{15} \\ \\ = \: &1 + {( - 1)}^{5} + {( - 1)}^{10} + {( - 1)}^{15} \\ \\& \quad [ \because {i}^{2} = \sqrt{ - 1} \times \sqrt{ - 1} = - 1 ] \\ \\ = \: &1 + {( - 1)}^{5} + { \big[ {( - 1)}^{2} \big]}^{5} + { \big[ {( - 1)}^{3} \big]}^{5} \\ \\ = \: &1 + {( - 1)}^{5} + {(1)}^{5} {( - 1)}^{5} \\ \\ = \: &1 + ( - 1) +(1) + ( - 1) \\ \\ = \: & \cancel1 - \cancel1 + \cancel1 - \cancel1 \\ \\ = \:&0 \\ \\ = \: & \mathbb{RHS} \end{aligned}$

Hence , Proved !

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.... Thank You !!!

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Answer 2

Step-by-step explanation:

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