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Question:
The simplified form of $i^{n} + i^{n +1} + i^{n +2} + i^{n +3}$ is
The simplified form of $i^{n} + i^{n +1} + i^{n +2} + i^{n +3}$ is
Updated On: Apr 17, 2024
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The Correct Option is A
Solution and Explanation
We have, $ i^{n} +i^{n+1}+i^{n+2}+i^{n+3} $
$=i^{n}\left(1+i+i^{2}+i^{3}\right) $
$=i^{n}(1+i-1-i)$
$\left[\because i^{2}=-1 \text { and } i^{3}=-i\right] $
$=i^{n} \times 0=0$
$=i^{n}\left(1+i+i^{2}+i^{3}\right) $
$=i^{n}(1+i-1-i)$
$\left[\because i^{2}=-1 \text { and } i^{3}=-i\right] $
$=i^{n} \times 0=0$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
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