query: Why does $ 1+2+3+\cdots+p = {(1⁄2)}\cdots(p+1) $
positive_passages.docid: 2675
positive_passages.text: Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
negative_passages.docid: 99003
negative_passages.text: How do we calculate this sum $\sum_{n=1}^{\infty} \frac{1}{n(n+1)\cdots(n+p)}$?
negative_passages.docid: 93948
negative_passages.text: Is my proof correct? $p_1p_2p_3\cdots p_n+1)$ cannot be the square of an integer
negative_passages.docid: 127205
negative_passages.text: Explain proof of irreducibility of $x^{p-1} + 2x^{p-2}+ \dots +(p-1)x + p$
negative_passages.docid: 111298
negative_passages.text: Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$
negative_passages.docid: 7201
negative_passages.text: Prove that $x_1^2+x_2^2+x_3^2=1$ yields $ \sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4} $
negative_passages.docid: 78309
negative_passages.text: On an expansion of $(1+a+a^2+\cdots+a^n)^2$
negative_passages.docid: 34430
negative_passages.text: Why does $(\frac{p-1}{2}!)^2 = (-1)^{\frac{p+1}{2}}$ mod $p$?
negative_passages.docid: 119759
negative_passages.text: Help ending a proof using binomial theorem: $\sum_{k=0}^n {n \choose k}\cdot\frac{(-1)^k}{k+1} = \frac{1}{n+1}$
negative_passages.docid: 33314
negative_passages.text: How many non-negative integer solutions does $x_1+x_2+\cdots+x_n=A$ have?
negative_passages.docid: 33633
negative_passages.text: Re:$ (a,b,c,d)=(pr+qs, qr-ps,pr-qs,ps+qr) $
negative_passages.docid: 27922
negative_passages.text: prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$
negative_passages.docid: 151646
negative_passages.text: Prove by induction that $\sum_{i=1}^{n} i \cdot 2^i = (n-1) \cdot 2^{n+1} +2$. Help finding my mistake
negative_passages.docid: 116356
negative_passages.text: Combinatorial proof of $2 \cdot 3^0 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots 2 \cdot 3^{n-1} = 3^n-1$
negative_passages.docid: 41645
negative_passages.text: Checking my understanding: $1 - 1 + 1 - 1 + 1 - ... = \frac{1}{2}$
negative_passages.docid: 9435
negative_passages.text: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.
negative_passages.docid: 118024
negative_passages.text: Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$