# Tagged Questions

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### Show that $f^{[n]}(0)=0$ for all $n=0,1,2…$

Let $$f(x)=\left\{ {\matrix{ {{e^{ - {1 \over {{x^2}}}}},x \ne 0} \cr {0,x = 0} \cr } } \right.$$ Show that $f^{[n]}(0)=0$ for all $n=0,1,2,\cdots$ The proof: First note that for ...
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### Proof of the inequality $F_i<(5/3)^i$ for the Fibonacci numbers

The example states: As an example, we prove that the Fibonacci numbers, F0 = 1, F1 = 1, F2 = 2, F3 = 3, F4 = 5,..., Fi = Fi - 1 + Fi - 2, satisfy Fi < (5/3)i, for all i >= 1. To do this, we ...
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### Prove that $133$ divides $11^{n+1} + 12^{2n−1}$ for all $n > 1$? [duplicate]

I have tried this question so hard but still stuck here. It seems like easily provable if all $n$ are all positive numbers but in this question, the $n$ is bigger than $1$. original question : prove ...
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### Proof by induction and inequalities

I am stuck on this question: given $a_1a_2≤(\frac{a_1+a_2}{2})^2$ prove by induction of m that $$a_1a_2...a_p≤(\frac{a_1+a_2+...+a_p}{p})^p$$ where $a_i$ are all positive and real and $p=2^m$ (an ...
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### Proof by Induction for Fundamental Thm of Arithmetic

Use induction to make our proof of the Fundamental Theorem of Arithmetic more rigorous. Recall that $p$ is prime iff for all $a,b\in\mathbb Z:p\mid(ab)$ implies $p\mid a$ or $p\mid b$. Prove that ...
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### Use induction to prove that $2^n$ divides $(2n)!$ for any $n\in\mathbb{Z}_{\geq0}.$ [duplicate]

Use induction to prove that $2^n$ divides $(2n)!$ for any $n\in\mathbb{Z}_{\geq0}.$ How would you do the inductive step for this proof? I have the base case done.
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### Fibonacci Proof with Induction [duplicate]

$$f(n) = \left\{\begin{matrix} 0 & n=1\\ 1 & n=2\\ f_{n-1} + f_{n-2} & n\geqslant 2\end{matrix}\right.$$ How can I prove by induction that $$f_{n} \geq \left ( 1.5 \right )^{n-1}$$ ...
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### Proof by induction: $2^n > n$

Base is $2^1 > 1$. Now we assume $2^n > n$ and try to obtain $2^{n+1} > (n+1)$. If I can use $2^n > 1$, I could just add that to $2^n > n$ and get $2^{n+1} > (n+1)$ but I don't ...
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### Can I use Strong Induction to prove graph theory? - Hamilton Path/Tournament Graphs

I am working on the below proof. I am still new to proves so I am wondering if you guys can answer the following questions: 1) Did I use the inductive hypothesis correctly here? (By the inductive ...
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### is it wrong to do this to solve an induction question

When doing an induction problem is it wrong to simply add the next variable to both sides? for example for all natural numbers $$4+9+14+19....+(5n-1)=\frac{n}{2}(3+5n)$$ assume true for k ...
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### Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n})$

Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n})$ I tried induction theorem, when $n = 1$ it is obviously right. But, say $n=k$, It does not make sense since I cannot ...
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### prove by induction: $3 + 5 + 7 + … + (2n+1) = n(n+2)$

Use the principle of mathematical induction to prove that $$3 + 5 + 7 + ... + (2n+1) = n(n+2)$$ for all n in $\mathbb N$. I have a problem with induction. If anyone can give me a little insight ...
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### Proving by induction propositions of the type $P(n_1, n_2, …, n_k)$, where $n_1, n_2, …,$ and $n_k$ are natural numbers

For example: I've seen proofs of the multinomial theorem that use induction in the number of terms that are elevated at some power, but none that use induction in the exponent instead of using it in ...
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### Proving a summation involving binomial coefficients.

I need to prove the following inductively: (http://upload.wikimedia.org/math/9/e/5/9e57871ba17c1ad48e01beb7e1bb3bb9.png) $$\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}$$ And for the life of me I can't ...
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### Validity of a proof by induction

By intuition, I would say that if L1 is a subset of L and that L is regular, then L1 is also regular, because L1 has less states than L2 and therefore there must be an automata for L1 too. However, ...
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### Proving that $fib(n) < (5/3)^n$ for $n \ge 1$ by induction

I know this has been shown before here but no post really answered my question. I had this problem given to me as an induction practice problem and I couldn't solve it without help. When I got the ...
We're currently learning about induction in my real analysis course. I came up with the following proof that is obviously false but cannot quite figure out why it fails. Here it is... "Any set $S$ of ...
### Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.
I am confused as to how to solve this question. For the Base case $n=1$, $(2^{2(1)} - 1)\,/\, 3 = 1$, base case holds My induction hypothesis is: Assume $2^{2k} -1$ is divisible by $3$ when $k$ is a ...