query: Proof without using induction
positive_passages.docid: 20271
positive_passages.text: Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction
negative_passages.docid: 45914
negative_passages.text: Proving that $n!≤((n+1)/2)^n$ by induction
negative_passages.docid: 2197
negative_passages.text: Induction principle for $n!<n^n$
negative_passages.docid: 51877
negative_passages.text: Why is this 'Proof' by induction not valid?
negative_passages.docid: 125220
negative_passages.text: Solving binomial theorem via induction
negative_passages.docid: 11867
negative_passages.text: Prove by induction that $n < 2^n$ for all $n \ge 1$
negative_passages.docid: 62613
negative_passages.text: Need to proof by induction (calculus1 n and n+1)
negative_passages.docid: 28541
negative_passages.text: How to prove the mathematical induction is true?
negative_passages.docid: 33526
negative_passages.text: How to prove this beautiful statement on linear forms using induction?
negative_passages.docid: 62177
negative_passages.text: Prove that: $2^n < n!$ Using Induction
negative_passages.docid: 107810
negative_passages.text: Induction with Binomial Coefficients
negative_passages.docid: 95000
negative_passages.text: Using induction to prove that $2 \mid (n^2 − n)$ for $n\geq 1$
negative_passages.docid: 71918
negative_passages.text: Mathematical Induction
negative_passages.docid: 135509
negative_passages.text: Proof writing: how to write a clear induction proof?
negative_passages.docid: 123757
negative_passages.text: Proving a summation inequality with induction
negative_passages.docid: 104492
negative_passages.text: Why is complete strong induction a valid proof method and not need to explicitly proof the base cases?
negative_passages.docid: 122998
negative_passages.text: number of edges induction proof