1
vote
1answer
35 views

Question about written proof for geometric summation

Suppose $\alpha$ $\ne$ $\beta$ $\in \{0, 2\}^\mathbb{N}$ Prove that $$\sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \ne \sum\limits_{k = 0}^\infty\frac{\beta(k)}{3^k}. $$ This is the written proof ...
2
votes
1answer
50 views

Question about proof with geometric sums

I am confused on how to write proofs for geometric sums. I think that using the well ordering principle to find the least n $\in$ $\mathbb{N}$ with $\alpha(n)$ $\ne$ $\beta(n)$ would be a good ...
1
vote
2answers
30 views

Proving that mean KDR in a videogame is one

This is not related to schoolwork. A friend of mine challenged me to prove that the mean KDR (assuming players can only die at the hands of other players) must always be equal to one. I have gotten ...
0
votes
0answers
30 views

Proof sum of permutation

I'm trying to prove: $$P(N) = \sum permutation(A,N)=1 \tag{1}$$ for the particular choice of the set $A = \{ \mu_1, \dots, \mu_n, 1-\mu_1, \dots, 1-\mu_n \}$, where $i = 1, \dots, N$ . So for ...
0
votes
3answers
400 views

Sum of $k {n \choose k}$ is $n2^{n-1}$

Proof that $\suṃ̣_{k=1}^{n}k {n \choose k}$ for $n \in \mathbb N$ is equal to $n2^{n-1}$. As a hint I got that $k {n \choose k} = n {n-1\choose k-1} $. I tried solving this by induction but, in the ...
0
votes
1answer
26 views

Fundamental Matrix with Sums

Let $$\Phi(t)=\begin{bmatrix} x_{11}(t) & x_{12}(t)\\ x_{21}(t) & x_{22}(t) \end{bmatrix} $$ be a fundamental matrix for $$x'=A(t)$$ where $$A=\begin{bmatrix} a_{11}(t) & a_{12}(t)\\ ...
1
vote
2answers
59 views

Prove $r^n - s^n = (r-s)\sum_{j=0}^{n-1} r^js^{n-j-1}$ by induction

Prove $$r^n - s^n = (r-s)\sum_{j=0}^{n-1} r^js^{n-j-1} $$ $(1)$by induction. I've verified that $$n=1: r^1 - s^1 = (r-s)(r^0s{1-0-1}) = r-s$$ Assume $(1)$ is true for $n \le k$. That is $$r^k - s^k ...
1
vote
1answer
42 views

How can I prove $ \sum_{i=0}^{k-1}x_{i}2^{i}\leq\sum_{i=0}^{k-1}2^{i}=2^{k}-1<2^{k} $

How can I prove this inequality property? Assumption: The values of $x_i$ can either be 0 or 1
4
votes
4answers
631 views

Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$ [duplicate]

Prove that $$\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}.$$ This problem is in the chapter about discrete random variables, but I have no idea what to go about substituting. I can't ...
2
votes
0answers
58 views

Is my proof correct? (Also formally)

Hello dear community! I just worked on a problem in my discrete mathematics text book and wondered if my approach to a specific exercise is correct. There are no solutions to it, that's the reason I ...
4
votes
6answers
97 views

Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?

How can I show that $$ \sum_{i=1}^{n-1}i=\binom{n}{2}? $$ This is what I have tried, but I do not know if it is correct: Proof. Let $n=2$. Then, $$ \begin{align} \sum_{i=1}^{1}i&=1\text{, ...
3
votes
0answers
207 views

Proving identity involving sum

I'm stuck trying to prove the following identity: $$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose j}(4S-i-k-3)!(4S-2i-1)}{(4S-i-j-1)!} $$ $$=\frac{(-1)^{j+k}(4S-2k-3)!{k+1 ...
2
votes
3answers
85 views

Problem with Proof of Inequality with Squares by Induction

I am a bit new to logical induction, so I apologize if this question is a bit basic. I tried proving this by induction: $$\left(\sum_{k=1}^nk\right)^2\ge\sum_{k=1}^nk^2$$ Starting with the base ...
3
votes
3answers
128 views

Proving that $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots + \frac{1}{n(n+1)} = \frac{n}{n+1}$

How would we go about proving that $$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots +\frac{1}{n(n+1)} = \frac{n}{n+1}$$
3
votes
4answers
1k views

Proving the sum of the first $n$ natural numbers by induction

I am currently studying proving by induction but I am faced with a problem. I need to solve by induction the following question. $$1+2+3+\ldots+n=\frac{1}{2}n(n+1)$$ for all $n > 1$. Any ...
2
votes
5answers
1k views

Proof via Induction for A Summation

I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically: $$ ...
4
votes
4answers
175 views

Help in proving $ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum\limits_{k=0}^{n} \frac{1}{k!} < 3 $.

I am trying to prove this statement for all $ n \geq 1 $ using induction: $$ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum_{k=0}^{n} \frac{1}{k!} < 3. $$ I said: Base case $ n = 1 $: $$ \left( ...