Evaluate the following i ^30+i^40+i^50+i^60
Answers
Answer:
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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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