query: Find out value of $(3Z^{100}+\frac{2}{Z^{100}}+1)(Z^{100}+\frac{2}{Z^{100}}+3)?$
positive_passages.docid: 142076
positive_passages.text: $c$ is a complex number that satisyfing $(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$
negative_passages.docid: 41792
negative_passages.text: If $abc=1$, then $\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $
negative_passages.docid: 57670
negative_passages.text: Show that $|z_1 + z_2|^2 < (1+C)|z_1|^2 + \left(1 + \frac{1}{C}\right) |z_2|^2$
negative_passages.docid: 44562
negative_passages.text: If $\left|z+\frac{2}{z}\right |=2$, then prove that max value of $|z|$ is $\sqrt 3 + 1$ (Corrected version)
negative_passages.docid: 68725
negative_passages.text: Find expression for : $ S_n =\sum_{i=1}^{n} \frac{i}{i^4+i^2+1} $
negative_passages.docid: 147665
negative_passages.text: $a_1=3;\ a_{n+1}=3^{a_n}$; find $a_{2004}\bmod100$
negative_passages.docid: 57659
negative_passages.text: Evaluate $\int_{\left | z \right |=3} (e^{z}-1)dz/(z(z-1)(z-i))$
negative_passages.docid: 49028
negative_passages.text: Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+$
negative_passages.docid: 30307
negative_passages.text: Find the value of $k$ in $\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}} \ge \sqrt{a}+\sqrt{b}+\sqrt{c}+k$
negative_passages.docid: 17168
negative_passages.text: Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$
negative_passages.docid: 117046
negative_passages.text: Show that $\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}}+\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}}+\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}$
negative_passages.docid: 32866
negative_passages.text: How find this sum $\sum\limits_{n=0}^{\infty}\frac{1}{(3n+1)(3n+2)(3n+3)}$
negative_passages.docid: 89698
negative_passages.text: Please help with $\sum\limits_{r=1}^{N}\frac{2r+3}{3^r(r+1)}$
negative_passages.docid: 3733
negative_passages.text: If $11z^{10}+10iz^9+10iz-11 = 0$. Then possible value of $\mid z \mid,$ is
negative_passages.docid: 35556
negative_passages.text: $f(z)=\frac{z}{1+|z|} $ Is 1-1?
negative_passages.docid: 116014
negative_passages.text: Write $\frac {1}{1+z^2}$ as a power series centered at $z_0=1$
negative_passages.docid: 58133
negative_passages.text: ${a^2}{\mid}({b^3} + 1)\;$and$\;{b^2}{\mid}({a^3} + 1)$