11 questions linked to/from Fake induction proofs
129 views

### Examples of wrong proof by induction [duplicate]

Could someone provide a wrong clime and its wrong proof by induction so that only the first step be wrong?
17k views

I just remembered about a problem/paradox I read years ago in the fun section of the newspaper, which has had me wondering often times. The problem is as follows: A maths teacher says to the class ...
I'm bugged by the following that's summarized on p. 109 of this PDF. False theorem: All horses are the same color. Proof by induction: $\fbox{$P(n)$is the statement: In every set of ... 5answers 5k views ### Prove by induction that$n^3 + 11n$is divisible by$6$for every positive integer$n$. Prove by induction that$n^3 + 11n$is divisible by$6$for every positive integer$n$. I've started by letting$P(n) = n^3+11nP(1)=12$(divisible by 6, so$P(1)$is true.) Assume$P(k)=k^3+11k$... 1answer 685 views ### Finding the error in this induction proof [duplicate] Claim: If$n$belongs to$\mathbb{N}$, and$p$and$q$are natural numbers with maximum$n$, then$p=q$. Let$S$be the subset of the natural numbers for which the claim is true.$1$belongs to$S$, ... 1answer 255 views ### Mathematical Induction. Horses made me question my understanding [duplicate] I recently read about the false inductive proof that all horses are the same colour. There are some mathSE threads about this already (MathSE_thread_1, MathSE_thread_2). After reading this, I now ... 3answers 115 views ### Proving$10\cdot n=0$for all$n\in\mathbb{Z}$with$n\geq 0$using strong induction The question says$10\cdot n=0$for all$n\in\mathbb{Z}$with$n\geq 0$. Here is my proof by strong induction: Base case:$10\cdot0=0$. Let$k\geq 0$, and suppose that for any$m\leq k$we have ... 2answers 97 views ### How does$( 1 - (1- \frac{1}{2^{2^k}}))$become$(1+ \frac{1}{2^{2^k}})$? How does$\left( 1 - \left(1- \frac{1}{2^{2^k}}\right)\right)$become$\left(1+ \frac{1}{2^{2^k}}\right)$? I distributed the former but got negative$-\frac{1}{2^{2^k}}$. So it does not match the ... 2answers 92 views ### Evaluate$\lim_{n\to\infty}\left(\sqrt{\frac{9^n}{n^2}+\frac{3^n}{5n}+2}-\frac{3^n}{n}\right)$Find, if it exists, the following limit:$\displaystyle\lim_{n\to\infty}\left(\sqrt{\frac{9^n}{n^2}+\frac{3^n}{5n}+2}-\frac{3^n}{n}\right)$. 3answers 98 views ### Proving${\sum}^n _{i=1}i = \frac {n(n+1)}{2}$by induction I am having problems understanding how to 'prove' a summation formula. I have the equation:$ {\sum}^n _{i=1}i = \frac {n(n+1)}{2} $Basis Step when:$ n=1  {\sum}^1 _{i=1}i = \frac {1(1+1)}{...
I retrieved an old math book and I'm delighted to share following exercise. The pencils in a box of crayons always have the same color. Proof by induction on the number $n$ of pencils in the box: ...