query: Induction Proof $4^k < 5^k$
positive_passages.docid: 153123
positive_passages.text: Proving by induction $4^n  0$
negative_passages.docid: 145274
negative_passages.text: Prove by induction $\sum \frac {1}{2^n} < 1$
negative_passages.docid: 90789
negative_passages.text: Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$
negative_passages.docid: 26912
negative_passages.text: Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$
negative_passages.docid: 67527
negative_passages.text: Proof by Induction that $3^n ≥ 1+2^n$
negative_passages.docid: 134000
negative_passages.text: Proof by Strong Induction: $n = 2^a b,\, b\,$ odd, every natural a product of an odd and a power of 2
negative_passages.docid: 45914
negative_passages.text: Proving that $n!≤((n+1)/2)^n$ by induction
negative_passages.docid: 8609
negative_passages.text: Prove by induction that $n!>2^n$
negative_passages.docid: 30783
negative_passages.text: Prove inequality $n<3^n$ using mathematical induction
negative_passages.docid: 88144
negative_passages.text: Induction Proof $n! < n^n$
negative_passages.docid: 2197
negative_passages.text: Induction principle for $n!<n^n$
negative_passages.docid: 66358
negative_passages.text: Prove by mathematical induction that $2n ≤ 2^n$, for all integer $n≥1$?
negative_passages.docid: 11867
negative_passages.text: Prove by induction that $n < 2^n$ for all $n \ge 1$
negative_passages.docid: 85731
negative_passages.text: Prove by induction that $n! > n^2$
negative_passages.docid: 96903
negative_passages.text: How to show via inductive proof (and without the use of the binomial theorem) that $ {n\choose k} \in \mathbb{N} $?
negative_passages.docid: 125297
negative_passages.text: Induction to prove $2n + 3 < 2^n$
negative_passages.docid: 51243
negative_passages.text: Proving a Statement using Mathematical Induction