query: Why does $1+2+\dots+2003=\dfrac{2004\cdot2003}2$?
positive_passages.docid: 2675
positive_passages.text: Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
negative_passages.docid: 148883
negative_passages.text: Sum of the series $\frac{1}{2\cdot 4}+\frac{1\cdot3}{2\cdot4\cdot6}+\dots$
negative_passages.docid: 29602
negative_passages.text: Integral $ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $
negative_passages.docid: 46401
negative_passages.text: Value of $\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}}$
negative_passages.docid: 60411
negative_passages.text: Why is $\frac{987654321}{123456789} = 8.0000000729?!$
negative_passages.docid: 17243
negative_passages.text: Remainder when $2009^{2009}-1982^{2009}-1972^{2009}+1945^{2009}$ is divided by $1998$
negative_passages.docid: 145444
negative_passages.text: Solve $\dfrac{1}{1+\frac{1}{1+\ddots}}$
negative_passages.docid: 1002
negative_passages.text: Where is the flaw in this "proof" that 1=2? (Derivative of repeated addition)
negative_passages.docid: 9156
negative_passages.text: Can someone explain this: 0.2 + 0.1 = 0.30000000000000004?
negative_passages.docid: 114308
negative_passages.text: Why does round(2.49999999999999992) returns 3
negative_passages.docid: 19962
negative_passages.text: Why is $\infty \cdot 0$ not clearly equal to $0$?
negative_passages.docid: 10400
negative_passages.text: Why adding these two double does not give correct answer?
negative_passages.docid: 115295
negative_passages.text: Why are the two dot product definitions equal?
negative_passages.docid: 41645
negative_passages.text: Checking my understanding: $1 - 1 + 1 - 1 + 1 - ... = \frac{1}{2}$
negative_passages.docid: 65028
negative_passages.text: Evaluate $\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx $
negative_passages.docid: 147635
negative_passages.text: Find the sum $\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots+\frac{1}{99\cdot 100}$
negative_passages.docid: 110262
negative_passages.text: Finding a closed formula for $1\cdot2\cdot3\cdots k +\dots + n(n+1)(n+2)\cdots(k+n-1)$