Question

please explain this question​

Answer 1

Answer:

This is the magic of i and 4.

The answer is 0.

Step-by-step explanation:

Check the attachment for your answer

Hope this will help you.

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Answer 2

Step-by-step explanation:

We have,

$\sum_{r = 1}^{4n + 7} i^{r}$

$= {i}^{1} + {i}^{2} + {i}^{3} + .... + {i}^{4n + 7}$

$= \frac{i(1 - {i}^{(4n + 7)}) }{1 - i}$

$= \frac{i(1 - {i}^{4(n + 1) + 3}) }{1 - i}$

$= \frac{i(1 + i)}{(1 - i)}$

$= \frac{i(1 + i)^{2} }{1 - {i}^{2} }$

$= \frac{i(1 + {i}^{2} + 2i) }{2}$

$= \frac{2 {i}^{2} }{2}$

$= {i}^{2} = - 1$