If a + 1/a = √3, then what is the value of a³⁰ + a²⁴ + a⁶ + 1?

Question: If a + 1/a = √3, then what is the value of a³⁰ + a²⁴ + a⁶ + 1?

Solution:
Given, a + 1/a = √3
Now,
a³ + 1/a³ = (a + 1/a)³ - 3 . a . (1/a)(a + 1/a)
⇒ a³ + 1/a³ = (√3)³ - 3(√3) [∵ a + 1/a = √3]
⇒ a³ + 1/a³ = 3(√3) - 3(√3)
⇒ a³ + 1/a³ = 0
⇒ a⁶ + 1 = 0 [Multiplying both sides by a³]

Then,
a³⁰ + a²⁴ + a⁶ + 1
= a²⁴ (a⁶ + 1) + (a⁶ + 1)
= (a²⁴ × 0) + 0
= 0