Question

For any integer k, i^(4k+1)

Answer 1

Answer:

∣α

k

−α

k+1

∣ represents the side length of a 14 sided polygon (tetradecagon) inscribed in a unit circle.

Similarly, ∣α

4k−1

−α

4k−2

∣ represents the side length of a 14 sided polygon inscribed in a unit circle.

Let l be the length of the side.

Hence, the required ratio =

3l

12l

=4

Answer 2

$\sf For \: any \: intege r k \: , \: \: {i}^{(4k + 1)} = i$

Given :

The expression

$\sf {i}^{(4k + 1)}$

To find :

The value of the expression

Solution :

Step 1 of 2 :

Write down the given expression

The given expression is

$\sf {i}^{(4k + 1)}$

Step 2 of 2 :

Find the value of the expression

$\sf {i}^{(4k + 1)}$

$\sf = {i}^{4k } . {i}^{1}$

$\sf = {( {i}^{2})}^{2k} . \: i$

$\sf = {( - 1)}^{2k} . \: i$

$\sf =1 \times i$

$\sf = i$

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