The value of the expression 2 (1+ 1 ) (1+ 1 ^ 2 )+3 (2+ 1 ) (2+ 1 ^ 2 ) +4 (3+ 1 ) (3+ 1…
Mathematics · JEE Advanced · NTA Exams — Complex Numbers and Quadratic Equations
The value of the expression
\(2\left(1+\frac{1}{\omega}\right)\left(1+\frac{1}{\omega^{2}}\right)+3\left(2+\frac{1}{\omega}\right)\left(2+\frac{1}{\omega^{2}}\right)\)
\(+4\left(3+\frac{1}{\omega}\right)\left(3+\frac{1}{\omega^{2}}\right)+\ldots \ldots+(n+1)\left(n+\frac{1}{\omega}\right)\left(n+\frac{1}{\omega^{2}}\right),\)
where ω is an imaginary cube root of unity, is
\(2\left(1+\frac{1}{\omega}\right)\left(1+\frac{1}{\omega^{2}}\right)+3\left(2+\frac{1}{\omega}\right)\left(2+\frac{1}{\omega^{2}}\right)\)
\(+4\left(3+\frac{1}{\omega}\right)\left(3+\frac{1}{\omega^{2}}\right)+\ldots \ldots+(n+1)\left(n+\frac{1}{\omega}\right)\left(n+\frac{1}{\omega^{2}}\right),\)
where ω is an imaginary cube root of unity, is
- \(\frac{\mathrm{n}\left(\mathrm{n}^{2}+2\right)}{3}\)
- \(\frac{\mathrm{n}\left(\mathrm{n}^{2}-2\right)}{3}\)
- \(\frac{\mathrm{n}^{2}(\mathrm{n}+1)^{2}+4 \mathrm{n}}{4}\)
- None of these
Answer
(C) n ^ 2 ( n +1)^ 2 +4 n 4
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