Question

If a⁴ + a³ + a² + a + 1 = 0, find the value of a¹⁰⁰ + a²⁰⁰ + a¹²⁵ + a¹²⁰ + a³⁰⁰.
1. 1
2. 3
3. 5
4. 6

Answer 1

Correct Answer - Option 3 : 5

Given:

a4 + a3 + a2 + a + 1 = 0

Formula used:

a^mn = (a^m)ⁿ

Calculation:

a4 + a3 + a2 + a + 1 = 0

⇒ a4 + a3 + a2 + a = – 1      ----(1)

Both sides multiply by a

[a × (a4 + a3 + a2 + a + 1)] = 0 × a

⇒ (a⁵ + a⁴ +  a³ + a² + a) = 0

From equation (1);

⇒ (a5 – 1) = 0

⇒ a5 = 1

According to the question:

a100 + a200 + a125 + a120 + a300

⇒ (a⁵)²⁰ + (a5)⁴⁰ + (a5)²⁵ + (a5)²⁴  + (a5)⁶⁰

⇒ (1)20 + (1)40 + (1)25 + (1)24  + (1)60

⇒ 1 + 1 + 1 + 1 + 1 = 5

∴ The answer is 5.