Questions tagged [binomial-theorem]
For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.
0
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1answer
22 views
Summation of series to n terms in trigonometry of complex numbers
The question says that: Sum the series
I have solved the answer as follows:
As the above picture, I don’t know what should I do after the step.
The question asks to solve the problem using ...
1
vote
1answer
43 views
Is my proof of $\binom{m+n}{r}=\binom{m}{0}\binom{n}{r} + \binom{m}{1}\binom{n}{r-1} + … + \binom{m}{r}\binom{n}{0}$ right?
As the title says, I was requested to prove
$\binom{m+n}{r}=\binom{m}{0}\binom{n}{r} + \binom{m}{1}\binom{n}{r-1} + ... +\binom{m}{r}\binom{n}{0}$
I was requested to do this using the following ...
4
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3answers
97 views
Help on proof of $\binom{n}{0}^2 + \binom{n}{1}^2 + … + \binom{n}{n}^2 = \binom{2n}{n}$
The proof is required to be made through the binomial theorem. I will expose the demonstration I was tought, and forward my questions after exposing it. You'll see question marks like this one (?-n) ...
1
vote
0answers
84 views
Summing Up Binomial Coefficients [duplicate]
$$
\frac{\sum_{r=0}^{24}\binom{100}{4r}\binom{100}{4r+2}}{\sum_{r=1}^{25}\binom{200}{8r-6}}
$$
I tried this problem using Complex number approach and arrived at the solution. When I tried the ...
2
votes
1answer
45 views
What is the intermediate step in this negative binomial proof?
On Slide 47 of these slides, there is a formula stating
$$ s_k(t) = - \sum_{j=0}^{k} s_j(0) \sum_{n=j}^{k} \frac{e^{-\lambda_n t} \prod_{m=j}^{k-1} \lambda_m }{\prod^{k}_{m=j, m \neq n} (\lambda_m - \...
1
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0answers
30 views
Simple Proof of Binomial Theorem for Negative Integer Powers
There's a vast amount of clutter on the internet about this which I've been trawling through but it does not answer exactly what I'm asking which, because superficially similar questions have been on ...
2
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2answers
57 views
To find the sum: $\frac {1}{n!} \sum \binom {n}{2+3r} x^{1+r}$
Sum the series:
$$
\frac {x}{2!(n-2)!}+\frac {x^2}{5!(n-5)!}+\frac {x^3}{8!(n-8)!}+....+\frac {x^{\frac {n}{3}}}{(n-1)!},
$$
$n$ being a multiple of $3$.(Math. Tripos, 1899)
My attempt
We may ...
0
votes
0answers
17 views
How to code for $\phi_{k,N,L}=\sum_{i=k-L+1}^{k}\frac{\phi_{i,N-1,L}}{(k-i)!}I_{[0,(N-1)(L-1)]}(i)$
I have the following equation
$$
\left[\sum_{k=0}^{m-1}\frac{1}{k!}x^k\right]^n=\sum_{k=0}^{n*(m-1)}\phi_{k,n,m}\times x^k.
$$
where $\phi_{k,N,L}$ function define by $$\phi_{k,N,L}=\sum_{i=k-L+1}^{...
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1answer
40 views
How to simplify this sum
I'm having a lot of issues with this question. Ive done problems where I need to find the coefficient or the constant using the binomial theorem but I'm not sure how to even begin doing this. I looked ...
2
votes
1answer
68 views
Using the binomial expansion to approximate a square root.
Find the first four terms of the expansion $\sqrt{1-4x}$ in ascending powers of x.
Hence, approximate $\sqrt{6}$ to four decimal places.
I've expanded it to form this: $1-2x-2x^2-4x^3$
When ...
0
votes
1answer
38 views
Binomial probability question!
How do I solve this question?
A school lab has sixteen computers. A teacher observes that, in the long run, in 80% of school days, at least 1 machine is not working properly. Assuming the probability ...
1
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1answer
52 views
Combinatorial identities, check the asymptotic behavior.
Check asymptotic of
C = $\sum_{k = 0}^{\frac{n}{2} - \sqrt{n}} k \binom{n}{k} = f(n) + O(g(n))$
In the beginning I tried to simplify the expression under the sum: $$k\binom{n}{k} = k \frac{n!}{k!(n-...
1
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1answer
89 views
Are there any power identities which don't belong to this list?
The problem of finding expansions of monomials, binomials etc. is classical and there is a lot of beautiful solutions have been found already, the most prominent examples are
Binomial Theorem, ...
3
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2answers
78 views
If $C_0, C_1, C_2, .., C_n$ are the binomial coefficients in the expansion of $(1+x)^n$
If $C_0, C_1, C_2,...,C_n$ are the binomial coefficients in the expansion of $(1+x)^n$, prove that:
$$C_{r}.C_{n} + C_{r+1}.C_{n-1} +......+ C_{n}.C_{r} = C(2n, n+r) =\dfrac {(2n)!}{(n-r)! (n+r)!}$$
...
-2
votes
1answer
31 views
Need help expanding this negative binomial expanison
I am an A level Student studying for my papers and this is a question from a past paper...the original equation is
$$\frac{1}{\sqrt{1+x}+\sqrt{1-x}}$$
which part a says to convert to the following ...
6
votes
1answer
88 views
Evaluate: $\sum_{j=0}^{k}\sum_{i=0}^{j}{k \choose j}^2{k \choose i}$
Can anybody help me to evaluate this sum $(1)$?
$$\sum_{j=0}^{k}\sum_{i=0}^{j}{k \choose j}^2{k \choose i}\tag1$$
I have manage to figure out:
$$\sum_{j=0}^{k}\sum_{i=0}^{j}{k \choose j}{k \choose ...
1
vote
3answers
162 views
Number of ways to express sum.
Consider three sets of cards colored Blue, Red and Yellow. Each set has cards numbered $1-10$. The $4$ remaining cards are all indistinguishable cards numbered $0$.
Card numbered $i$ has the ...
2
votes
4answers
75 views
coefficient $x ^ n$ in development
The advisor asks to verify that the coefficient of
$$x^n$$
in the development of:
$$(1+x)^{2n}+x(1+x)^{2n−1}+x2(1+x)^{2n−2}+......+x^n(1+x)^n$$
is equal to
$$\binom{2n+1}{n}$$
I tried for summations ...
1
vote
1answer
34 views
Proof of the following combinatorial identity
Prove
$$\sum_{m\le i\le k-l} \binom{k-i-1}{l-1} \binom{i-1}{m-1} = \binom{k-1}{m+l-1}$$
where $m$, $l$, $i$, $k$ are positive integers and $k\ge m+l$.
Is this related to Vandermonde's identity?
1
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3answers
71 views
Proving $\sum_{m=0}^n\binom{n}{m}^2 \binom{m}{n-k}=\binom{n}{k}\binom{n+k}{k}$
How can I prove this?
$$\sum_{m=0}^n\binom{n}{m}^2 \binom{m}{n-k}=\binom{n}{k}\binom{n+k}{k}$$
$$ 0\le k \le n $$
I developed the expressions, but they are not the same. I do not know if it will be ...
0
votes
1answer
36 views
Why is this equality with sums and monomials true?
Consider $$(x_1 + ... + x_n)^k = \sum_{|\alpha| = k}c_{\alpha}x^{\alpha}$$ where $x^{\alpha} = x_{1}^{a_1}\cdots x_{n}^{a_n}$ and $|\alpha| = a_1 + ... + a_n$.
Why is this true? Is it something to do ...
1
vote
7answers
82 views
How to find divisibility of a very large number
I was asked this question in a test:
The number $111111...111$ ($1$ comes 91 times) is a:
A) Prime number
B) Composite Number
C) divisible by $\frac{10^7 -1}{9}$
The answer is B ...
2
votes
0answers
58 views
strange binomial coefficient identity
Can someone please show why the following holds:
$$ \sum_{k=0}^n\binom{n-1}{k-1}p^{k}q^{n-k} = p \sum_{k=1}^n\binom{n-1}{k-1}p^{k-1}q^{n-k} $$
Thanks in advance.
Edit:
Thank you for commenting.
...
3
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1answer
43 views
Combinatorial proof for $\sum_{i=k}^n (2i-k) \binom{i-1}{k-1}^2 = k \binom{n}{k}^2$
Give combinatorial proof for:
$\sum_{i=k}^n (2i-k) \binom{i-1}{k-1}^2 = k \binom{n}{k}^2$
RHS: We want to make sequence A and B with their element only 0 and 1, and the number of element 1 is k in A ...
0
votes
1answer
63 views
Find $a^3+b^3+c^3-3abc$ (binomial theorem) [duplicate]
$$a=\sum_{n=0}^\infty\frac{x^{3n}}{(3n)!}\\b=\sum_{n=1}^\infty\frac{x^{3n-2}}{(3n-2)!}\\c=\sum_{n=1}^\infty\frac{x^{3n-1}}{(3n-1)!}$$Find $a^3+b^3+c^3-3abc$:
$(a)\ 1$
$(b)\ 0$
$(c)-1$
...
1
vote
1answer
50 views
Combinatorial Proof (Argument)
Give a combinatorial proof for:
$$\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^{k} \binom{n-1-j}{k-j} 2^j$$
2
votes
4answers
65 views
Show that $\lim [(2n)^\frac{1}{n})] = 1$
Prove that $\lim ((2n)^\frac{1}{n}) = 1$.
I have obtained the following:
$$(2n)^\frac{1}{n} = 1 + k_{n}; n > 1$$
$$(2n) = (1 + k_{n})^n$$
By the Binomial Theorem:
$$(1 + k_{n})^n = 1 + k_{n} + \...
1
vote
3answers
42 views
Show that $\sum_{r=0}^{n} \binom{n}{r} r p^r q^{n-r} = np$, given $p+q=1$
How can I prove
$$\sum_{r=0}^{n} \binom{n}{r} {r} p^r q^{n-r} = np,$$
given that $p+q=1$?
I think it is about applying
$$(1+x)^n=\sum_{r=0}^{n} \binom{n}{r} x^r.$$
Although I am quite unsure about ...
1
vote
1answer
48 views
Domain limitations on generating function for Legendre polynomials
The generating function for legendre polynomials is $$\frac{1}{\sqrt(1-2ut+u^2)}=\sum_{i=0}^\infty u^iP_i(t)$$ where $P_i$ is $i^{th}$ legendre polynomial however for binomial expansion of left hand ...
0
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2answers
31 views
Binomial series expansion of a trinomial?
In electrostatics, the potential of a charge $q$ placed on the $z$-axis at $z=a$ is
\begin{equation}
\phi=\frac{1}{4\pi \epsilon_0}\frac{q}{r_1}
\end{equation}
where $r_1$ is the distance from the ...
2
votes
0answers
31 views
$x^{2}+x+1$ is a factor of $(x+1)^{n} - x^{n} - 1$, whenever [duplicate]
Q) $x^{2}+x+1$ is a factor of $(x+1)^{n} - x^{n} - 1$, whenever
(A) $n$ is odd
(B) $n$ is odd and a multiple of $3$
(C) $n$ is an even multiple of $3$
(D) $n$ is odd and not a multiple of $3$
...
1
vote
1answer
47 views
Closed form of the sum of two binomial expansions
I would like to know if there is a simple closed form for the following expression:
$(x+y)^n + (x+z)^n$
Expanding the above I get
$(y^n + z^n) + nx(y^{n-1}+z^{n-1}) + \frac{(n-1)n}{2}x^2(y^{n-2} + ...
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votes
3answers
26 views
Binomial Expansion based on equation for evaluation
I have this question that has really stumped me, it is supposed to be done via Binomial kind of expansion.
If $x+\frac1x=10$ find the value of $x^3+\frac1{x^3}$.
So I hope some one has an approach ...
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2answers
39 views
Newtons Binomial Theorem identity
I have seen a lot of identites being discussed here but I still haven't seen the one I'm having a problem with.
I need to conclude that $\sum\limits_{i=1}^{n} i\binom{n}{i} \frac{(-4)^{i-1}}{5^{i-n}} ...
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votes
3answers
77 views
If $(1+x)^{n}=C_{0} + C_{1} x + C_{2} x^2+…C_{n} x^n$
If $(1+x)^{n}=C_{0} + C_{1} x + C_{2} x^2+......C_{n} x^n$, prove that:
$$\textrm {a}. C_{0}C_{n}+C_{1}C_{n-1}+..........+C_{n}C_{0}=\dfrac {{2n}!}{{n}! \cdot {n}!}$$
$$\textrm {b}. {C_{0}}^{2}+{C_{1}...
4
votes
1answer
88 views
$\int\frac{dx}{x(x+1)(x+2)\cdot\space…\space\cdot(x+n)}$ [duplicate]
I've been trying to solve explicitly the following indefinite integral:
$$\int\frac{dx}{x(x+1)(x+2)\cdot\space...\space\cdot(x+n)}$$
I tried to perform partial fraction decomposition, and after ...
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2answers
41 views
Showing an Inequality Holds True
If $x$ is a positive real number and $n$ is a positive integer, prove the inequality, $\sqrt[n]{1+x}-1 \le \frac{x}{n}$.
I tried to show this was true by doing the following:
For $n = 1$, $\sqrt[1]{...
1
vote
1answer
42 views
How to evaluate a binomial sum with $2n$ in the exponent.
The question is to evaluate the sum $$\sum_{k=0}^n\binom nk 3^{2n-k}$$
have tried fitting into the binomial form of $\binom nk \times x^k\times y^{n-k}$
but I can't seem to bring it to the correct ...
1
vote
3answers
44 views
For which $k$ the following equation has the greatest value: $ k \cdot\binom{99}{k} $?
After some manipulation I got $$\frac{99!}{(k-1)!\cdot(99-k)!} $$
So I guess I have to find $k$ for which I get the smallest denominator, but I don't know where to go from there.
1
vote
1answer
22 views
Maximum value of coefficient in Multinomial Expansion
Find the maximum value of coefficient in the expansion of $(x+y+z+w)^{25}$.
Basically what the question is saying is that all term will be of type $k x^{r_1}y^{r_2}z^{r_3}w^{r_4}$ so what can be ...
1
vote
0answers
82 views
Finding a sequence with some summation properties $\sum\binom{n}{k}a_k=\binom{2n}{n}$
I was wondering if there exists a nontrivial sequence of numbers $a_{k}$ with the following property:
$$\sum_{k=0}^{N}\binom{n}{k}a_{k}=\binom{2n}{n},$$
where the integer $N=\lfloor\tfrac12n\rfloor$ ...
10
votes
1answer
132 views
Prove that $16 ^ {2023} + 1$ is divisible by $17 ^ 2$.
Prove that $16 ^ {2023} + 1$ is divisible by $17 ^ 2$.
It is clear that $16 ^ {2023} + 1$ is divisible by $17$, but why it is divisible by $17 ^ 2$ is not clear.
4
votes
6answers
177 views
Simple proof that $(1 + 1/n)^n$, $n \in \mathbb N$, is bounded above?
Here is the context of the problem. I'd like to prove that the sequence
$$a_n = \left( 1 + \frac 1n \right)^n$$ converges using the monotone convergence theorem. It is straightforward using Bernoulli'...
2
votes
2answers
172 views
Why does the Binomial Theorem use combinations and not permutations for its coefficients?
I have been trying to understand the Binomial Theorem formula. I can see that it works.
What I don’t understand is how or why using combinations finds the coefficients.
What I mean is, isn’t each ...
5
votes
3answers
253 views
Question related to Pascal Triangles and Counting Patterns
I was working on two separate problems and came across a potential pattern involving Pascal's triangle. I would like to know if Pascal's triangle applies to either of these problems and if so, is ...
0
votes
3answers
101 views
In the expansion of $a{x}^3 {(2+ax)}^{11}$ , the coefficient of the term in $x^5$ is $11880$. Find the value of $a$ .
This is a question from the ib question bank which has the answer however the method isnt explained well enough. Its binomial theorem. Can someone explain the method in detail of how to deal with ...
3
votes
1answer
39 views
Proving the connection between the Binomial Theorem and the product rule for derivatives
Let $a(x)$ and $b(x)$ be smooth functions, i.e they are infinitely times differentiable.
I have made the assumption that the derivative for the function
$$f(x)= (a\cdot b)(x)$$
can be given by
$$...
1
vote
3answers
91 views
How can I find the coefficient of $x^6$ in $(1+x+\frac{x^2}{2})^{10}$ efficiently with combinatorics?
To find the coefficient of $x^6$ in $(1+x+\frac{x^2}{2})^{10}$,
I used factorization on $(1+x+\frac{x^2}{2})$ to obtain $\frac{((x+(1+i))(x+(1-i)))}{2}$, then simplified the question to finding the ...
8
votes
3answers
159 views
How can I prove $\frac {d}{dx} {x^n} = n x^{n-1}$ for $ n \in \Bbb R$ without circular reasoning? [duplicate]
I just cannot prove that $$\frac {d}{dx} {x^n} = n x^{n-1}$$ for $ n \in \Bbb R$.
For $n \in \Bbb{N}$, I can use the definition of a derivative :
$$\frac {d}{dx}x^n = \lim_{h \rightarrow 0} \frac{(...
0
votes
2answers
41 views
Simplifying an expression with binomial coefficients and powers of $2$
$$\binom{n}{0} \cdot 2^n + \binom{n}{1} \cdot 2^{n-1} + \binom{n}{2} \cdot 2^{n-2} + \dots + \binom{n}{n} \cdot 2^{n-n}$$
Anyway to simplify this such that it can become of 'closed' form (i.e. a ...
