Question

Evaluate the following i ^30+i^40+i^50+i^60​

Answer 1

Answer:

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Answer 2

Answer:

0

Concept Used:

Values of some iⁿ:

• i⁰ = 1

• i = √-1 = i

• i² = -1

• i³ = -i

• i⁴ = 1

• i⁻¹ = -i

• i⁻² = -1

• i⁻³ = i

• i⁻⁴ = 1

Finding the value of iⁿ:

(i) CASE I [n > 4]

In order to compute iⁿ for n > 4 , we divide n by 4 and obtain the remainder r.Let m be the quotient when n is divided by 4. Then,

n = 4m + r

n = 4m + r, where 0 ≤ r < 4

⇒ iⁿ = i⁴ᵐ ⁺ ʳ = (i⁴)ᵐ × iʳ = 1 × iʳ = iʳ [∵ i⁴ = 1]

⇒ iⁿ = iʳ

(ii) CASE II

i⁰ = 1

(iii) CASE III [n < 4]

In order to compute i⁻ⁿ for n < 4 , we divide n by 4 and obtain the remainder r.Let m be the quotient when n is divided by 4. Then,

n = 4m + r

n = 4m + r, where 0 ≤ r < 4

⇒ i⁻ⁿ = i⁻⁽⁴ᵐ ⁺ ʳ⁾ = (i⁻⁴)ᵐ × i⁻ʳ = 1 × i⁻ʳ = i⁻ʳ [∵ i⁻⁴ = 1]

⇒ i⁻ⁿ = i⁻ʳ

Step-by-step explanation:

i³⁰ + i⁴⁰ + i⁵⁰ + i⁶⁰

= i⁽⁴*⁷ ⁺ ²⁾ +  i⁽⁴*¹⁰ ⁺ ⁰⁾ + i⁽⁴*¹² ⁺ ²⁾ + i⁽⁴*¹⁵ ⁺ ⁰⁾

= i² + i⁰ + i² + i ⁰

= -1 + 1 + (-1) + 1

= -1 + 1 - 1 + 1

= 0

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