Answer :
the sum [tex]\( \sum_{n=1}^5 \left(\frac{1}{i}\right)^n \)[/tex] is equal to -i. So, the correct option is D.
Let's evaluate the series [tex]\( \sum_{n=1}^5 \left(\frac{1}{i}\right)^n \)[/tex]:
[tex]\[ \sum_{n=1}^5 \left(\frac{1}{i}\right)^n = \left(\frac{1}{i}\right)^1 + \left(\frac{1}{i}\right)^2 + \left(\frac{1}{i}\right)^3 + \left(\frac{1}{i}\right)^4 + \left(\frac{1}{i}\right)^5 \][/tex]
Simplify each term:
[tex]\[ = \frac{1}{i} + \frac{1}{-1} + \frac{1}{-i} + \frac{1}{1} + \frac{1}{i} \][/tex]
Combine like terms:
[tex]\[ = \frac{1}{-i} + \frac{1}{1} + \frac{1}{i} \][/tex]
Now, to add these fractions, we need a common denominator. The common denominator is [tex]\( i \)[/tex], so:
[tex]\[ = \frac{-i}{-i \cdot i} + \frac{i}{i \cdot i} + \frac{1}{i \cdot -i} \][/tex]
Combine the numerators:
[tex]\[ = \frac{-i + i - 1}{-i^2} \][/tex]
Simplify the numerator and replace [tex]\( i^2 \)[/tex] with -1:
[tex]\[ = \frac{-1}{-(-1)} \][/tex]
Finally:
[tex]\[ = \frac{-1}{1} \][/tex]
So, the sum [tex]\( \sum_{n=1}^5 \left(\frac{1}{i}\right)^n \)[/tex] is equal to -1. Therefore, the correct option is:
[tex]\[ \text{d. -i} \][/tex]
