Question

find the value of the following expression
​

Answer 1

$\underline{ \huge\bf\red{QuestiOn} :}$

Find the value of the expression :

$\boxed{1 + {i}^{2} + {i}^{4} + {i}^{6} + {i}^{8} + ... + {i}^{20} }$

$\underline{ \: \huge\bf\green{SolutiOn} \: :}$

We have,

$\begin{aligned} &\quad 1 + {i}^{2} + {i}^{4} + {i}^{6} + {i}^{8} + ... + {i}^{20} \\ \\ & = 1 + {( - 1)}^{2} + {( - 1)}^{4} + {( - 1)}^{6} + {( - 1)}^{8} + ... + {( - 1)}^{20} \\ \\ & = 1 + ( - 1) + ( + 1) + ( - 1) + ( + 1) \\ & \qquad + ( - 1) + ( + 1) + ( - 1) + ( + 1) \\ & \qquad + ( - 1) + ( + 1) \\ \\ & = 1 + 5( + 1) + 5( - 1) \\ \\ & = 1 + \cancel{5} - \cancel{5} \\ \\ & = \bf 1 \end{aligned}$

$\red{\rule{5.8cm}{0.02cm}}$

$\mid \underline{\underline{\LARGE\bf\green{Brainliest \: Answer}}}\mid$

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Answer 2

$\underline{ \huge\bf\red{QuestiOn} :}$

Find the value of the expression :

$\boxed{1 + {i}^{2} + {i}^{4} + {i}^{6} + {i}^{8} + ... + {i}^{20} }$

$\underline{ \: \huge\bf\green{SolutiOn} \: :}$

We have,

$\begin{gathered}\begin{aligned} &\quad 1 + {i}^{2} + {i}^{4} + {i}^{6} + {i}^{8} + ... + {i}^{20} \\ \\ & = 1 + {( - 1)}^{2} + {( - 1)}^{4} + {( - 1)}^{6} + {( - 1)}^{8} + ... + {( - 1)}^{20} \\ \\ & = 1 + ( - 1) + ( + 1) + ( - 1) + ( + 1) \\ & \qquad + ( - 1) + ( + 1) + ( - 1) + ( + 1) \\ & \qquad + ( - 1) + ( + 1) \\ \\ & = 1 + 5( + 1) + 5( - 1) \\ \\ & = 1 + \cancel{5} - \cancel{5} \\ \\ & = \bf 1 \end{aligned}\end{gathered}$

$\red{\rule{5.8cm}{0.02cm}}$

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