Prove that (1 + i10 + i20 + i30) is a real number.

1.	Prove that (1 + i10 + i20 + i30) is a real number.
Answer» L.H.S = (1 + i¹⁰+ i²⁰ + i³⁰) = (i + i^{4 x 2 + 2} + i^{4 x 5} + i^{4 x 7 + 2}) Since ⇒ i⁴ⁿ = 1 ⇒ i⁴ⁿ⁺¹ = i ⇒ i⁴ⁿ⁺² = -1 ⇒ i⁴ⁿ⁺³ = -1 = 1 + i² + 1 + i² = 1 + -1 +1 + -1 = 0, which is a real no. Hence, (1 + i¹⁰ + i²⁰ + i³⁰) is a real number.

Answer»

L.H.S = (1 + i¹⁰+ i²⁰ + i³⁰)

= (i + i^{4 x 2 + 2} + i^{4 x 5} + i^{4 x 7 + 2})

Since ⇒ i⁴ⁿ = 1

⇒ i⁴ⁿ⁺¹ = i

⇒ i⁴ⁿ⁺² = -1

⇒ i⁴ⁿ⁺³ = -1

= 1 + i² + 1 + i²

= 1 + -1 +1 + -1

= 0, which is a real no.

Hence, (1 + i¹⁰ + i²⁰ + i³⁰) is a real number.

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