Question

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if (i = sqrt{-1}), simplify the expression (i^{3}+i^{4}+i^{5}+i^{6}+i^{7}+i^{8}+i^{9}) into one of the values listed below:
1
-1
0
-i
i

Explanation:

Step1: Recall powers of i

$i=\sqrt{- 1}$, $i^{2}=-1$, $i^{3}=i^{2}\cdot i=-i$, $i^{4}=(i^{2})^{2}=(-1)^{2}=1$, $i^{5}=i^{4}\cdot i = i$, $i^{6}=i^{4}\cdot i^{2}=1\times(-1)= - 1$, $i^{7}=i^{4}\cdot i^{3}=1\times(-i)=-i$, $i^{8}=(i^{4})^{2}=1^{2}=1$, $i^{9}=i^{8}\cdot i = i$.

Step2: Substitute into the expression

$i^{3}+i^{4}+i^{5}+i^{6}+i^{7}+i^{8}+i^{9}=-i + 1+i-1 - i+1+i$.

Step3: Combine like - terms

$(-i + i - i+i)+(1-1 + 1)=1$.

Answer:

1