query: Proof by Induction: $ \sum_{i=1}^{n} i \cdot i! = (n+1)! -1$
positive_passages.docid: 70611
positive_passages.text: Prove by Mathematical Induction: $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$
negative_passages.docid: 96332
negative_passages.text: Prove by induction that $\sqrt[n+1]{n+1}<\sqrt[n]{n}$ for $n\ge 3$
negative_passages.docid: 88292
negative_passages.text: Prove by induction that $\sum\limits_{i=1}^n \frac{1}{n+i} \leq \frac{3}{4}$
negative_passages.docid: 111515
negative_passages.text: Proving that $-a=(-1)\cdot a$
negative_passages.docid: 106888
negative_passages.text: Factorial identity $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+\cdots$
negative_passages.docid: 33903
negative_passages.text: Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
negative_passages.docid: 62556
negative_passages.text: Prove $3|n(n+1)(n+2)$ by induction
negative_passages.docid: 91197
negative_passages.text: Proving $n! \ge 2^{n-1 }$for all $n\ge1 $by mathematical Induction
negative_passages.docid: 22666
negative_passages.text: Proving that if $A^n = 0$, then $I - A$ is invertible and $(I - A)^{-1} = I + A + \cdots + A^{n-1}$
negative_passages.docid: 48333
negative_passages.text: Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$
negative_passages.docid: 88055
negative_passages.text: Use induction to prove the following: $1! + 2! + .... + n! \le (n + 1)!$
negative_passages.docid: 34299
negative_passages.text: Combinatorial proof of $\sum_{k=0}^n k \cdot k! = (n+1)! -1$
negative_passages.docid: 26901
negative_passages.text: induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$
negative_passages.docid: 101899
negative_passages.text: Proof By Induction of a Sum
negative_passages.docid: 36639
negative_passages.text: Prove by induction that $\sum_{r=0}^{n}\begin{pmatrix}n \\ r \end{pmatrix}=2^k$
negative_passages.docid: 16177
negative_passages.text: Proof by Induction: $\sum_0^nx^i=(1-x^{n+1})/(1-x)$
negative_passages.docid: 25054
negative_passages.text: Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ by Induction