Question

The value of 1 + i² + i⁴ + i⁶ + ... + i⁴⁰ (where i = √-1) is:

1. i
2. 0
3. 1
4.

-1

Answer 1

Correct Answer - Option 3 : 1

Concept:

The imaginary unit i is defined as i = √-1.

i^{4n + 1} = i, i4n + 2 = -1, i4n + 3 = -i, i4n = 1, etc.

 

Calcultion:

The given expression can be written as:

i⁰ + i2 + i4 + i6 + ... + i40

= (i⁰ + i4 + i⁸ + ... + i40) + (i2 + i6 + i10 + ... + i38)

= [1 + 1 + ... (11 times)] + [-1 + -1 + ... (10 times)]

= 11 - 10

= 1.