Question

The value of the expression 1.(2 – ω) + (2 – ω²) + 2.(3 – ω)(3 – ω²) + ....... + (n – 1)(n – ω)(n – ω²), where ω is an imaginary cube root of unity is ______.

विकल्प

• `[("n"("n" + 1))/2]^2`

• `[("n"("n" + 1))/2]^2 - "n"`

• `[("n"("n" + 1))/2]^2 + "n"`

• `("n"^2("n" + 1)^2 - "n")/2`

Answer 1

The value of the expression 1.(2 – ω) + (2 – ω²) + 2.(3 – ω)(3 – ω²) + ....... + (n – 1)(n – ω)(n – ω²), where ω is an imaginary cube root of unity is `underlinebb([(n(n + 1))/2]^2 - n`.

Explanation:

General term (r – 1)(r – ω)(r – ω²)

(r – 1){r² – rω² – rω + ω³}

(r – 1){r² – r(ω + ω²) + ω³}

(r – 1)(1 + r + r²) = r³ – 1

`sum_("r" = 1)^"n"("r"^3 - 1) = sum_("r" = 1)^"n" "r"^3 - sum_("r" = 1)^"n" 1 = [("n"("n" + 1))/2]^2 - "n"`