Q85 / 180 · COMEDK 2023 (Morning Shift) · Mathematics
If $i=\sqrt{-1}$ and $n$ is a positive integer, then $i^n+i^{n+1}+i^{n+2}+i^{n+3}$ is equal to
- A$i^n$
- Correct answerB0
- C1
- D$i$
Solution
The expression is $i^n + i^{n+1} + i^{n+2} + i^{n+3}$.
Factoring out $i^n$, we get $i^n(1 + i + i^2 + i^3)$.
Since $i^2 = -1$ and $i^3 = -i$, the expression becomes $i^n(1 + i - 1 - i)$.
Simplifying the terms inside the parentheses: $1 - 1 + i - i = 0$.
Therefore, $i^n(0) = 0$.
Answer: 0
Factoring out $i^n$, we get $i^n(1 + i + i^2 + i^3)$.
Since $i^2 = -1$ and $i^3 = -i$, the expression becomes $i^n(1 + i - 1 - i)$.
Simplifying the terms inside the parentheses: $1 - 1 + i - i = 0$.
Therefore, $i^n(0) = 0$.
Answer: 0
Hence, the correct answer is option: 'B'.