Exponents

Imaginary number rules,

To the power of -

1: i

2: -1

3: -i

4: 1

5: i

and so on.

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Now we have to find how many times each will occur throughout the sequence.

Since 96th term will be 1, this means that each power of i outcome will occur 24 times. So we have 24i-24-24i+24i

97th term will be i

98th term will be -1

99th term will be -i

So add the three above to the expression we had earlier to get : 25i-25-25i+24i

Simplifies to -25+24i

\(\boxed{-25+24i}\)

Simplify \[i^1+i^2+i^3+\cdots+ i^{97} + i^{98}+i^{99}.\]
\(\large i^1+i^2+i^3+\cdots+ i^{97} + i^{98}+i^{99}\).

 \(\begin{array}{|rcll|} \hline && \displaystyle i^1+i^2+i^3+i^4+i^5+\cdots+ i^{97} + i^{98}+i^{99} \\\\ &=& (i^1+i^2+i^3+i^4)(1+i^4+i^8+i^{12}+\cdots+ i^{96} )-i^{100} \\\\ &=& (i^1+i^2)(1+i^2)(1+i^4+i^8+i^{12}+\cdots+ i^{96} )-i^{100} \quad | \quad i^2 = -1 \\\\ &=& (i^1+i^2)(1-1)(1+i^4+i^8+i^{12}+\cdots+ i^{96} )-i^{100} \\\\ &=& (i^1+i^2)(\underbrace{1-1}_{=0})(1+i^4+i^8+i^{12}+\cdots+ i^{96} )-i^{100} \\\\ &=& 0-i^{100} \\\\ &=& -(i^2)^{50} \quad | \quad i^2 = -1 \\\\ &=& -(-1)^{50} \\\\ &=& -1 \\ \hline \end{array}\)