Question

`i^0 +i^1+i^2+i^3+............+i^(2021) = ?`

Include work.

Answer 1

1+i

Explanation:

I do believe i to be the imaginary unit.

Let's write out some partial sums from power=0 to power=7 or whatever we need to see a pattern.

i^0=1

i^0+i^1=1+i

i^0+i^1+i^2=1+i+-1=i

i^0+i^1+i^2+i^3=i+i^3=i+-i=0

i^0+i^1+i^2+i^3+i^4=0+i^4=0+1=1

Hmmm.... we might see 1+i, then i, then 0 again.... let's see.

i^0+i^1+i^2+i^3+i^4+i^5=1+i

Coolness so we should see a pattern

Sum from power=0 to power=multiples of 4 will give us 1.

Sum from power=0 to power=remainder of 1 when final power is divided by 4 gives us 1+i.

Sum from power=0 to power=remainder of 2 when final power is divided by 4 gives us i.

Sum from power=0 to power=remainder of 3 when final power is divided by 4 gives us 1

0.

So 2021 divided by 4....

Since 2020 is a multiple of 4, then 2021 has a remainder of 1 when divided by 4.

So the answer is 1+i.