query: In mathematical induction, why can we say if it is true for n=1, n=k and n=k+1 then it is true for all numbers?
positive_passages.docid: 50268
positive_passages.text: Why is mathematical induction a valid proof technique?
negative_passages.docid: 94601
negative_passages.text: Mathematical Induction Question About A Basis Step
negative_passages.docid: 136844
negative_passages.text: Mathematical induction question: why can we "assume $P(k)$ holds"?
negative_passages.docid: 59280
negative_passages.text: Need help with Proof by Strong Induction question
negative_passages.docid: 69943
negative_passages.text: why is ${n+1\choose k} = {n\choose k} + {n\choose k-1}$?
negative_passages.docid: 51877
negative_passages.text: Why is this 'Proof' by induction not valid?
negative_passages.docid: 31890
negative_passages.text: Prove using induction that from a set of $n+1$ numbers from $\{1,2,..2n\}$, at least one number will evenly divide another.
negative_passages.docid: 41784
negative_passages.text: Prove using mathematical induction that $n < 3 ^n$ for all positive integers n.
negative_passages.docid: 123427
negative_passages.text: Proving the sum of the first $n$ natural numbers by induction
negative_passages.docid: 92418
negative_passages.text: Prove by induction Fibonacci equality
negative_passages.docid: 87941
negative_passages.text: Prove by mathematical induction that $\forall n \in \mathbb{N} : \sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k} $
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: 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: 68480
negative_passages.text: Proof by Induction: Prove that $2^n > n^2$, for all natural numbers greater than or equal to $5$
negative_passages.docid: 33757
negative_passages.text: For what natural numbers is $n^3 < 2^n$? Prove by induction
negative_passages.docid: 81136
negative_passages.text: Prove by induction that... $1+3+5+7+...+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$
negative_passages.docid: 72026
negative_passages.text: Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $