Tagged Questions

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Help understanding about Series and Sequences

I was hoping to see if anyone could help me out about explaining about Sequences and Series? Because I am getting bit stuck on how to really understand the concept of certain Sequences Here's an ...
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What are the total number of ordered permutations of a list of elements (possibly repeated)?

This question is a part of a TopCoder problem. I am learning algorithms, and just got stuck at this (not homework). Suppose we have a set $A$ of integer elements, such that $n(A) = a$ (number of ...
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Geometric series for values between 0 and 1

I am given that geometric series is defined as the following $1-x+x^2-x^3+x^4$ for values in range $0<x<1$. I am also told expected value can be calculated by using the following equation: ...
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Prove boundedness of recurrence relation

For a number sequence $\{y_n\}$ we know that $y_{n+1} = 2y_n-y^2_n$ If: $0<y_0<1$ show that $0<y_n<1$ for all integers $n>0$ I've tried solving the recurrence relation, but I couldn't ...
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Discrete Math sequence question

The question is find $a3$: $a_0 = 2, a_1 = 4$ and $a_{k+2} = 3a_{k+1}-a_k$ for any integer $k \geq 0$ I know the answer is 26, although how do you get the answer?
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How to find the sum of sequence $1+4+4^2+\cdots+4^{X+Y}$?

I see the following sequence and it's: $$h=1+4+4^2+\cdots+4^{X+Y}=\frac{4^{X+Y+1}-1}{4-1}$$ how we get this sequence? I know this is a primary question but I confused :)
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Double summation index problem

I often meet the following situation: $$\sum\limits_{n=0} ^\infty \sum\limits_{k=0} ^n f(k)g(n-k)=\sum\limits_{p=0} ^\infty \sum\limits_{q=0}^\infty f(p)g(q)$$ While intuitively this is very clear ...
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How do you derive the continuous analog of the discrete sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …$?

I was wondering what the rate of growth of the sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...$$ was, and found the related question, $n$th term of $1, 2, 2, 3, 3, 3, 4, 4 ,4, 4, 5...$, in which one of ...
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I'm searching for the formula of the series $\sum_{n=0}^{\infty}a^{n^l}$

I'm searching for the sum-formula (if exists) of the following power series: $$\sum_{n=0}^{\infty}a^{n^l}$$ where $l=2,3,....$, and $|a|<1$.
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Recurrence problem for $a_5$

Assume that the sequence $\{a_0,a_1,a_2,\ldots\}$ satisﬁes the recurrence $a_{n+1} = a_n + 2a_{n−1}$. We know that $a_0 = 4$ and $a_2 = 13$. What is $a_5$?
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Compute $\sum_{i=0}^{2n} (-3)^i$ by splitting the series into two parts.

Compute $\sum_{i=0}^{2n} (-3)^i$ by splitting the series into two parts. How do I split it into two parts? All I can tell so far is that the sum is going to be a positive number (probably) because ...
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Compute the sum $\sum_{i=0}^n 5^{i+1}-5^i$

Compute the sum: $$\sum_{i=0}^n 5^{i+1}-5^i$$ with the hint, "start by writing out (and expanding) the sum." So I did and got $$4 + 20 + 100...$$ with the appearance of going to infinity. Is ...
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How do you use induction on a recursive sequence using different variables?

I've been working on some recursive sequences for my Discrete class. I've understood most of them, but I've come to a new question which I'm confused about. A sequence $C_{0}$, $C_{1}$, $C_{2}$ is ...
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geometric series used to work out big O notation for resizing an array in a stack

It's a geometric series $$1 + 2 + 4 + \cdots + 2^k = \frac{1 - 2^{k+1}}{1 - 2}$$ Here, $2^k$ = N. You get $1 + 2 + 4 + \cdots + N = \frac{1 - 2N}{-1}$. Therefore, $2 + 4 + \cdots + N = 2N−2$. When ...
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Infinite Sum with Combination

I am trying to figure out what the following sum converges to: $$\sum_{n=0}^\infty {6+n\choose n}x^n(6+n),\qquad\qquad0<x<1$$ An answer would be great, but if you have an explanation, that'd ...
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Find a closed form for the sum $∑(x^3 - 2x)$ from $x=1$ to any number $n$

Find a closed form for the sum $∑(x^3 - 2x)$ from $x=1$ to any number $n$. Can someone explain to me what a closed form is and how to approach this problem?
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What's the maximum deviation from loan amortization

Suppose you have a loan with principle P and fixed interest rate i compounded daily. Suppose you make fixed payments every month, but not on the same day. The only constraint is that you make every ...
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$if An \subseteq A$ for all $n \in \mathbb{N},$ then $\bigcup_{n=1}^\infty An \subseteq A$

I was given this as an exercise in my discrete math class and I have been having a lot of trouble, I am not really sure how to approach a problem like this. Any help is appreciated!Thank you! (this is ...
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List naturals in ascending product order

Define an ascending product ordering as a sequence $(x_1,y_1), (x_2,y_2), \ldots$ with the following properties: Each pair of naturals is represented: For any integers $a\ge b>0$, we know that ...
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number of indices differences

I have a counting problem below: Let $n>2$ be integer and $p>0$ be a real number. For all $1\leq i<j\leq n$, suppose $a_{ij}$'s and $b_k$'s satisfy $a_{ij}=b_{j-i}=p^{j-i}$ ...
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$\frac{1}{7} + \frac{1\cdot 3}{7\cdot 9} + \frac{1\cdot 3\cdot 5}{7\cdot 9\cdot 11} + …$

If the sum $\frac{1}{7} + \frac{1\cdot 3}{7\cdot 9} + \frac{1\cdot 3\cdot 5}{7\cdot 9\cdot 11} + ...$ to 20 terms is $\frac{m}{n}$, reduced fraction, then what is $n-4m$? This is a question I dug ...
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Uniqueness proof of binary representation

I'm having trouble understanding this proof of uniqueness of binary representation of integers: Suppose there exists an integer $n$ with two different binary representations. Let these be: ...
Help with sequence: $a_k = 5*3^k + 7*2^k$ - Induction
Let $a_k$ be a sequence, where $a_0 = 12$, $\;$ $a_2 = 29$ and $a_k = 5a_{k-1} - 6a_{k-2}$ , $k\geq 2$ . I need to prove, using induction, that $a_k = 5\times 3^k + 7\times 2^k \; , k\geq 0$ . ...
I am unable to solve the following recursive equation which I must solve in my research problem. Please give me advice or solution to the problem. For $K=\min(N/2,C)$ and N,C T_c, T_s,p,T are ...