All Questions
Tagged with summation algebra-precalculus
977
questions
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Proving $\sum_{1}^{n} \left\lceil\log_{2}\frac{2n}{2i-1}\right\rceil=2n -1 $
Show that $$\sum_{i=1}^{n} \left\lceil\log_{2}\frac{2n}{2i-1}\right\rceil=2n -1 $$ where $ \lceil\cdot\rceil$ denotes the ceiling function.
My method: one way would be observe each part of the ...
- 977
3
votes
2
answers
166
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Could someone help me to derive the closed form for this sum?
$$\sum_{r=1}^n \frac{(-1)^{r+1}(r+2)}{2^{r+1}}$$
I've already looked on Wolfram Alpha for the solution but I would like to know the step-by-step process to get there. I've attempted to make this into ...
- 205
0
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1
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89
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Is it possible to calculate how many digit number I have to write if I write $n^2$ sequence from 1 to $1000^2$ without using calculator?
From the fact that, for any positive integer n, it will require $1+\lfloor\log_{10}(n)\rfloor$ digits to write. When $\lfloor\ \rfloor$ is floor function.
If I write "1,4,9,16,25,...,1000000"...
- 725
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2
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56
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Rewriting $\sum_{{i,j=0}\:\:i\ne j}^n \binom{n}{i}\:\:\binom{n}{j}$ [duplicate]
Solve $$\sum_{{i,j=0}\:\:i\ne j}^n \binom{n}{i}\:\:\binom{n}{j}$$ This was a contest math problem which I was not able to solve.
My work: I was very unsure about how to approach this question. In my ...
- 860
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2
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88
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Finding maximum possible sum of numbers in arithmetic progression
I have a game where prices increment linearly depending on the amount you have, so if you have 0 widgets buying 1 costs \$5 and buying 2 costs \$15 (10 for the second), the $30 for 3 (15 for the third)...
- 163
4
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1
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57
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Show that these two sum forms are equal
I am trying to prove the following lemma (related to formal derivatives for reference):
Let $R$ be a ring, $k\in \mathbb{Z}^+ (\text{this means }k\geq 1)$, and $f_0,\dots,f_k,g_0,\dots,g_k\in R$. Then
...
- 548
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0
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86
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Limit of a sum and two ways yielding two answers
$$\lim_ {n\to \infty} \sum_{r=0}^n \frac{1}{(2r+1)(2r+3)(2r+2)}$$
Now, what I did here was to first break up the general term of the sum using partial fractions, yielding
$$ \frac{1}{(2r+1)(2r+2)(2r+3)...
- 33
1
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1
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64
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If $a_sk^s+a_{s-1}k^{s-1}+...+a_0$ is the basis representation of $n$ with respect to the basis $k$. Then, $0<n\leq k^{s+1}-1$.
If $a_sk^s+a_{s-1}k^{s-1}+...+a_0$ is the basis representation of $n$ with respect to the basis $k$. Then, $$0<n\leq k^{s+1}-1$$.
My attempt:-
By basis represantation, we know that $0\leq a_j<k,...
- 849
1
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3
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177
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calculate :$2000 \choose 2000$+....+$2000 \choose 8 $+$2000 \choose 5$+$2000 \choose 2$
my attempt:
let's put $2000 \choose 2000$+...+$2000 \choose 1001 $+...+$2000 \choose 8 $+$2000 \choose 5$+$2000 \choose 2$=$A+B$
with $A$=$2000 \choose 2000$+...+$2000 \choose 1001 $
and
$B$=$2000 \...
user1069990
1
vote
2
answers
124
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Changing summation indices
How does one prove the following result, where $x$ is a three-parameter function defined on $\mathbb Z^3$? $$ \Sigma_{\ell=1}^{P}\Sigma^{\ell-1}_{i=0} x(\ell,i,\ell-i) \quad = \quad \Sigma^{P}_{j=1}\...
- 1,139
1
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1
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70
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Is there a closed form of $x_2x_3\cdots x_n + x_1x_3\cdots x_n + \dots + x_1x_2\cdots x_{n-1}$?
Given real numbers $x_1,\dots,x_n \in \mathbb{R}$, does there exist a closed form for the expression
$$A_n := x_2x_3\cdots x_n + x_1x_3\cdots x_n + \dots + x_1x_2\cdots x_{n-1} = \sum_{i=1}^n \prod_{j=...
- 2,740
1
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0
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40
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Relating $\sum_{k=1}^N a_k^2 e^{\frac{2\pi i}{N}k}$ to $(\;\sum_{k=1}^N a_k e^{\frac{2\pi i}{N}k}\;)^2$
Consider the following expression
$$
\sum_{k=1}^N a_k^2 e^{\frac{2\pi i}{N}k}\tag{1}
$$
where $i$ is the imaginary number. How may I relate it to the following expression
$$
\left(\sum_{k=1}^N a_k e^{\...
- 3,435
0
votes
2
answers
49
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write the given summation in terms of $x^n$ instead of $x^{3n}$
I have $\sum_{n\geq0}(2n)x^{3n} =0+2x^3+4x^6+6x^9+...$ , but i want to write this summation in terms of $x^n$ instead of $x^{3n}$ .How can i do it ?
I thought that if i can write $n/3$ in place of $n'...
user1066985
2
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2
answers
120
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Product of $n$ terms of sequence where the $n^{th}$ term is of the form $(x^{a^n}+1)$
While practicing from a book I found a product in the form $$(x^{a^1}+1)\cdot(x^{a^2}+1)\cdot(x^{a^3}+1)\cdot(x^{a^4}+1)$$ and was immediately curious if I could a formula to solve the product for $n$ ...
- 121
0
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2
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86
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Evaluating $\sum_{k=0}^{\infty}\frac{1}{2k-1}$
$\sum_{k=0}^{\infty}\frac{1}{2k-1}$ is a convergent series.
Is there some way to evaluate $\sum_{k=0}^{\infty}\frac{1}{2k-1}$
This does not look like arithmetic or geometric series to me.
Please help
- 914
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2
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115
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$\frac{1^2}{1\cdot3} + \frac{2^2}{3\cdot5} + \frac{3^2}{5\cdot7}+\cdots+\frac{500^2}{999\cdot1001} = ?$
I found this problem in a high school text book.
Let $ \displaystyle s = \frac{1^2}{1\cdot3} + \frac{2^2}{3\cdot5} + \frac{3^2}{5\cdot7}+\cdots+\frac{500^2}{999\cdot1001}$. Find $s$.
How I tried:
...
- 479
1
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1
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109
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How to factorize and solve equations with $\Sigma$ notation?
I have a few doubts about the properties of sigma notation, $\Sigma$ . My questions rely on factorization and solving equations with $\Sigma$.On account of the fact that my questions are correlated, I ...
user1056027
1
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0
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48
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seperating two variables in a function with summation
I'm building a data analysis program that perform on big chunks of data, the issue I'm having is the speed of some operations; to be exact I have a function that takes two variables in this form : $$f(...
- 113
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votes
1
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46
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Find $\sum_{j = 1}^{2004} i^{2004 - F_j}$ where $F_n$ is the nth Fibonacci number
The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_n = F_{n - 1} + F_{n - 2}$ for $n \ge 3.$
Compute
$$\displaystyle\sum_{j = 1}^{2004} i^{2004 - F_j}.$$
I tried computing the first few ...
- 800
1
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1
answer
49
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How to simplify $\displaystyle\sum_{i=0}^{n-1}\displaystyle\sum_{j=i}^{n-1} (j-i+1)$
I am new to simplifying summations, and I am not sure what to do from the point I am at right now. Here is what I have done so far:
$\begin{alignat*}{2}
\sum_{i=0}^{n-1}\sum_{j=i}^{n-1} (j-i+1) &...
- 343
2
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3
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162
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Evaluating $\sum \limits_{k=1}^{\infty} \frac{\binom{4k}{2k}}{k^2 16^k}$
I want to find the closed form of:
$\displaystyle \tag*{}\sum \limits_{k=1}^{\infty} \frac{\binom{4k}{2k}}{k^2 16^k}$
I tried to use the taylor expansion of $\frac{1}{\sqrt{1-x}}$ and $\frac{1}{\sqrt{...
3
votes
2
answers
114
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Why $ \sum _{k=0} ^{n-1} \binom{n-1}{k} x^{k+1} = \sum _{k=1} ^{n} \binom{n-1}{k-1} x^k $?
I'm struggling to understand how to get from this:
$$ \sum _{k=0} ^{n-1} \binom{n-1}{k} x^{k+1} $$
to this:
$$ \sum _{k=1} ^{n} \binom{n-1}{k-1} x^k $$
I always have a problem understand the ...
1
vote
4
answers
115
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If $\Sigma_{k=1}^n \frac{1}{k(k+1)}= \frac{7}{8}$ then what is $n$ equal to?
If $S_n=\Sigma_{k=1}^n \frac{1}{k(k+1)}= \frac{7}{8}$ then what is $n$ equal to?
So, the most obvious course of action in my mind is to find a closed form for the partial summations, but alas, this ...
user637978
2
votes
3
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92
views
Expansion of $\prod_{j=1}^{n} \left( \sum_{i = 1}^{m} x_{i, j} \right)$
I would like to know if there is a sum-of-products expansion for the following product-of-sums. A special case is given here for the difference of two entries.
$$\prod_{j=1}^{n} \left( \sum_{i= 1}^{m} ...
- 1,876
0
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1
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39
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Summation of Binomial Coefficients Simplification [duplicate]
The following equality is given by an online solver, but I would like to understand how the simplification is made:
$\binom{A+1}{B+1} = \sum_{i = h}^{h + A - B} \binom{i}{h} \binom{A-i}{B-h}$
So far I ...
0
votes
1
answer
39
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Does anyone know how to bound decaying exponential series of the form $\exp(\sum_{k = i + 1}^n-C/(k+1))$
Consider a series with the form,
$$\exp(\sum_{k = i + 1}^n -C/(k+1))$$
where $C > 0$ is some constant and $n, i$ are integers with the assumption $n > i+1$
I wish to find some type upper-bound ...
- 557
0
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1
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65
views
Calculating the final sum of an investment with a specific daily growth of rate over a period of time.
Calculating the final sum of an investment with a specific daily growth of rate over a period of time.
I do apologize if this question is very basic for the vast majority of people in this forum but ...
- 3
1
vote
2
answers
58
views
How to calculate the following sum? $\sum_{1 \le i < j \le s} x_i \times x_j$
$\sum_{1 \le i < j \le s} x_i \times x_j$
The elements of the $x_1, x_2, ... , x_s$ sequence are: one $-1$, two $-2$'s, three $-3$'s, ..., n $-n$'s. We also have $s = \frac{n \times (n+1)}{2}$ ...
- 109
4
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3
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760
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Probability you end dice rolling sequence with 1-2-3 and odd total number of rolls
Here's a question from the AIME competition:
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an ...
- 1,415
2
votes
5
answers
147
views
Simplify $\sum n(n+1)$ [duplicate]
According to WolframAlpha,
$$\sum_{n=1}^k n(n+1)=\frac{1}{3}k(k+1)(k+2)$$
and it is easy to verify this if we use induction.
However, I would like to know how one can actually come up with this, other ...
- 1,358