query: How to simplify this equation: $1\cdot N + 2(N-1) + 3(N -2) + \cdots + i(N - i + 1) + \dots + N \cdot1$?
positive_passages.docid: 11612
positive_passages.text: How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
negative_passages.docid: 70611
negative_passages.text: Prove by Mathematical Induction: $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$
negative_passages.docid: 74294
negative_passages.text: How to simplify $\sum_{i=1}^{k}\binom{n + i - 1}{i}$?
negative_passages.docid: 69791
negative_passages.text: How to solve: T(n) = T(n - 1) + n
negative_passages.docid: 13789
negative_passages.text: solve $T(n)=T(n-1)+T(\frac{n}{2})+n$
negative_passages.docid: 111298
negative_passages.text: Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$
negative_passages.docid: 99843
negative_passages.text: How to simplify $ \sum_{k=0}^{\infty} \binom{k + r - 1}{k} x^k$
negative_passages.docid: 114212
negative_passages.text: How to use Mathematical Induction to prove $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{n(n + 1)} = \frac{n}{n + 1}$?
negative_passages.docid: 120430
negative_passages.text: A question about how to express a fraction as ${1\over q_1}+{1\over q_2}+ \cdots+{1\over q_N}$
negative_passages.docid: 30851
negative_passages.text: Find the summation $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \cdots$
negative_passages.docid: 62342
negative_passages.text: $x_1 \cdot \dots \cdot x_n=1 \implies x_1 + \dots + x_n \ge n$
negative_passages.docid: 115213
negative_passages.text: How to simplify $\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}$
negative_passages.docid: 74898
negative_passages.text: $1^n +2^n + \cdots +(p-1)^n \mod p =$?
negative_passages.docid: 51864
negative_passages.text: $\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \right) = -1$
negative_passages.docid: 109706
negative_passages.text: Fast way to come up with solutions to $x(x-1)(x-2)(x-3)=1$?
negative_passages.docid: 152113
negative_passages.text: Formula to this pattern? $1$, $11$, $21$, $1211$, $111221$, $\ldots$
negative_passages.docid: 7066
negative_passages.text: Calculate simple expression: $\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$