# Compute #i+i^2+i^3+\cdots+i^{258}+i^{259}#. ?

##### 2 Answers

#### Answer:

#### Explanation:

The trick is to know about the basic idea of sequences and series and also knowing how

The powers of

We can regroup

We know that

So:

We know that within each of these groups, every term is the same, so we are just counting how much of these are repeating.

From here on out, it's pretty simple. You just evaluate the expression:

According to the polynomial identity

$$

\sum_{k=0}^n x^k = \frac{x^{n+1}-1}{x-1}

$$

we have

$$

i+i^2+i^3+\cdots + i^{259} = \frac{i^{260}-1}{i-1}-1

$$

but

$$

i+i^2+i^3+\cdots + i^{259} = -1

$$