Q141 / 240 · KCET 2026 · Mathematics
$\sum\limits_{n=1}^{4} (-1)^{2n} \cdot i^{2n} = $
- A$2$
- B$-i$
- Correct answerC$0$
- D$i$
Solution
The given expression is $\sum\limits_{n=1}^{4} (-1)^{2n} \cdot i^{2n}$
Since $(-1)^{2n} = ((-1)^2)^n = 1^n = 1$ for all integers $n$, and $i^{2n} = (i^2)^n = (-1)^n$, the expression simplifies to:
$\sum\limits_{n=1}^{4} 1 \cdot (-1)^n = \sum\limits_{n=1}^{4} (-1)^n$
Expanding the sum for $n = 1, 2, 3, 4$:
$(-1)^1 + (-1)^2 + (-1)^3 + (-1)^4$
$-1 + 1 - 1 + 1 = 0$
Answer: $0$
Since $(-1)^{2n} = ((-1)^2)^n = 1^n = 1$ for all integers $n$, and $i^{2n} = (i^2)^n = (-1)^n$, the expression simplifies to:
$\sum\limits_{n=1}^{4} 1 \cdot (-1)^n = \sum\limits_{n=1}^{4} (-1)^n$
Expanding the sum for $n = 1, 2, 3, 4$:
$(-1)^1 + (-1)^2 + (-1)^3 + (-1)^4$
$-1 + 1 - 1 + 1 = 0$
Answer: $0$
Hence, the correct answer is option: 'C'.