Question

Solve this mcq ?
this is about series and sequesnces

Answer 1

the sum
`\( \sum_(n=1)^5 \left((1)/(i)\right)^n \)` is equal to -i. So, the correct option is D.

Let's evaluate the series
`\( \sum_(n=1)^5 \left((1)/(i)\right)^n \)`:

`\[ \sum_(n=1)^5 \left((1)/(i)\right)^n = \left((1)/(i)\right)^1 + \left((1)/(i)\right)^2 + \left((1)/(i)\right)^3 + \left((1)/(i)\right)^4 + \left((1)/(i)\right)^5 \]`

Simplify each term:

`\[ = (1)/(i) + (1)/(-1) + (1)/(-i) + (1)/(1) + (1)/(i) \]`

Combine like terms:

`\[ = (1)/(-i) + (1)/(1) + (1)/(i) \]`

Now, to add these fractions, we need a common denominator. The common denominator is
`\( i \)`, so:

`\[ = (-i)/(-i \cdot i) + (i)/(i \cdot i) + (1)/(i \cdot -i) \]`

Combine the numerators:

`\[ = (-i + i - 1)/(-i^2) \]`

Simplify the numerator and replace
`\( i^2 \)` with -1:

`\[ = (-1)/(-(-1)) \]`

Finally:

`\[ = (-1)/(1) \]`

So, the sum
`\( \sum_(n=1)^5 \left((1)/(i)\right)^n \)` is equal to -1. Therefore, the correct option is:

`\[ \text{d. -i} \]`