Question

Find the value of 1+i^2+i^3+i^4+....+i^20
a) 0
b) 1
c) 20
d) -1

Answer 1

The sum of the powers of i from 1 to 20 is 0 because the powers of i are cyclical with a period of 4, and every four terms in the sequence sum up to 0. Therefore, the entire sum of 1+i^2+i^3+i^4+....+i^20 equals 0.

Step-by-step explanation:

To find the value of the expression 1+i^2+i^3+i^4+....+i^20, we need to recognize the powers of the imaginary unit i. The powers of i are cyclical:

• i^1 = i
• i^2 = -1
• i^3 = -i
• i^4 = 1

So, every four powers of i, the sequence repeats itself. We can divide 20 by 4 to find that i^20 will be the same as i^(4*5), which is i^4 raised to the 5th power, and since i^4 = 1, i^20 is also 1.

Now the sum can be simplified by grouping every four terms:

1. (1 + i^2 + i^3 + i^4)
2. (i^5 + i^6 + i^7 + i^8)
3. ...
4. (i^17 + i^18 + i^19 + i^20)

Each group equals 0 (1-1+0). Therefore, the entire sum also equals 0.

The correct answer is a) 0.