# Tagged Questions

For questions about mathematical induction, a method of mathematical proof. Mathematical induction generally proceeds by proving a statement for some integer, called the *base case*, and then proving that if it holds for one integer then it holds for the next integer. This tag is primarily meant ...

289 views

### Number or regions formed when $n$ points on a circle are joined

The maximum number $R_{n}$ of regions formed when $n$ points on a circle are joined in pairs is $\frac{1}{24}\left(n^{4}-6n^{3}+23n^{2}-18n+24\right)$. This is a fact that I have read in several ...
74 views

### How to prove an identity containing binomial coefficients

I am trying to prove the identity $$\sum_{k=1}^n (3^k - 1) \binom{n}{k} = 4^n - 2^n$$ where $\binom{n}{k}$ is the binomial coefficient n over k or n choose k.
66 views

### Proving $n! \ge 2^{n-1 }$for all $n\ge1$by mathematical Induction

Im trying to solve the following question In the second step where do they get $k!=2^k-1?$
178 views

29 views

### How to generalize induction from this definition?

Definition: An inductive set $A$ is a one that satisfies: $1\in A$ and $k \in A\implies k+1\in A$. If we characterize the natural numbers as the set which has the following properties: $\Bbb N$ is ...
49 views

### Proof by Induction for Natural Numbers

Show that if the statement $$1 + 2 + 2^{2} + ... + 2^{n - 1} = 2^{n}$$ is assumed to be true for some $n,$ then it can be proved to be true for $n + 1.$ Is the statement true for all $n$? ...
55 views

### Use Proof of Induction to prove $\sum_{k=1}^{2n} (-1)^k k = n$

Base Case: \begin{eqnarray*} \sum_{k=1}^{2n} (-1)^k k = n\\ (-1)^1 (1) + (-1)^2(2) &=&1 \\ 1=1 \end{eqnarray*} Inductive Step: For this step we must prove that \begin{eqnarray*} ...
67 views

### Is my induction proof of $2^{n} > 2n+1$ correct?

Hello I am wondering if anyone can conform that the method I use in the following proof is valid. If not please inform me/ point me in the right direction. It is a very basic question, i.e. to prove ...
70 views

### inequality of two functions

I have the following problem. I need to show that $b(i)>b(i-1)^{i-1}$ for $i>k$ for some $k$. $b$ is the following function: $b(1)=2$ and $b(n)=2^{b(n-1)}$ I tried to do this by induction, but ...
Assume that $|x + y| \leq |x| + |y|$ for all $x,y \in {Z}.$ Use this assumption and induction to prove that $$|a_1 + a_2 + ... + a_n| \leq |a_1| + |a_2| + ... + |a_n|$$ for all integers $n \geq 2$ ...