If x⁵ + x⁴ + x³ + x² + x = 0 find x¹⁰⁰⁰ + x¹⁰⁰ + x¹⁰

3

x⁵ + x⁴ + x³ + x² + x = 0

This equation is in G.P.

x⁵ = first term

\(\frac{1}{x}\) = common ratio

then first term \(\frac{ { \left(Common ratio \right) }^{n} -1 }{(Common ratio - 1)}\)

⇒ x⁵\(\frac{ { \left({ \left(\frac{1}{x}\right) }^{5}-1 \right)}}{\frac{1}{x}-1}\) = 1

1 - x⁵ = 0

x⁵ = 1

Put in find

\( { \left(x^5\right) }^{200} \) + \( { \left(x^5\right) }^{20} \) + \( { \left(x^5\right) }^{2} \) = 1 +1 +1 = 3