For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.

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2answers
27 views

Odd Power terms of binomial theorem proof

I want to acquire all the terms of $(p+q)^n$ where the power of p is odd. Note that $p=1-q$ ($p$,$q$ probabilities) Ex. For $(p+q)^2=p^2+q^2+2pq$ I want to acquire only $2pq$(only term with odd ...
0
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4answers
52 views

Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$

Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$ I know that \begin{equation*} (1 + x)^n = 1 + nx +\frac {n(n-1)}2!\cdot x^2 +\frac {n(n-1)(n-2)}3! \cdot x^3 +... \end{equation*} ...
2
votes
1answer
43 views

Show that the binomial series satisfies: $(1+x)f'(x)=\alpha f(x)$

If $f(x) = \sum_{n=0}^{\infty}{\alpha\choose n}x^n$, show (without assuming the results of the binomial theorem) that $$(1+x)f'(x)=\alpha f(x)$$ for $|x|<1$ I've already shown that the sum ...
1
vote
1answer
20 views

Show that the following expansion is valid for first $n$ terms

The question is as follows, I've gone ahead and attempted as much as I could of it twice, however the final answer that I'm receiving does not match. I'm omitting the partial fractions steps; if ...
2
votes
2answers
49 views

Prove that $a^{p^2}\equiv a^p\pmod{p^2}$ for every $a\in\mathbb{Z}$

Question: Prove that $a^{p^2}\equiv a^p\pmod{p^2}$ for every $a\in\mathbb{Z}$ First note that when $p$ is prime and $1\leq r\leq p-1$ the binomial coefficient $$\binom{p}{r}=\frac{p!}{r!(p-r)!}$$ is ...
1
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0answers
16 views

Lower bounding the binomial summation

For $n>1$ and $~0<p < 1$, can we lower bound the following binomial series in terms of $n$ and $p$ $$\sum_{i=\lceil p n \rceil}^n {n \choose i} (p )^i(1-p)^{(n-i)}.$$ I have posted the ...
3
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2answers
80 views

Formal power series coefficient problem

Find the coefficient of: $[x^{33}](x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}$ I have figured out that I need to use this identity: $(1-x)^{-k} = \sum\limits_{i>=0} \binom {n+k-1} {k-1} x^n $ But I ...
1
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1answer
12 views

Solving product of binomial powers in integers

I have to find $a,b \in \mathbb{Z}^*$ such that $$(1 + x)^5(1+ax)^6 = 1 + bx + 10x^2 + \ldots + a^6x^{11}$$ Now, as far as I can tell for all $a$ there is a solution when $b = 5$, so a solution is ...
0
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1answer
27 views

Binomial Expansion

Let $f, g:[0,1]\rightarrow R$ be functions such that $$\left\{\begin{array} & f(x)=g(x)=0 & \Leftrightarrow x=1 \\ f(x), g(x) \neq 0 & \Leftrightarrow x \neq 1\end{array}\right.$$ Now ...
2
votes
4answers
78 views

$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $

$$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $$ (can this be duplicate? I think not) I tried it using many methods $1.$ Solve this conventionally taking $1^\infty$ form in no ...
0
votes
3answers
41 views

Summation Of Binomial & Factorial Series

Looking for an explicit formula for the following: $$ S = \sum _{i=j}^n \frac{\binom{i}{j}}{(i+1)!} $$ any Ideas?
0
votes
2answers
33 views

Determine a, b and n in the following expression (binomial theorem backwards)

You are given the following expression: $(ax + by)^{n} = -15120x^{4}y^{3}$ Determine the constants $a$, $b$ and $n$. My attempt to solve this problem is by trying to use the binomial theorem ...
1
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1answer
46 views

Binomial Theorem…extra indexed term.

I have the following expression: $$\sum_{i=0}^{n}\binom{n}{i}(2x+1)^{n-i}(-1)^ii!$$ Without the $i!$, the above expression would simply reduce to $(2x)^n$, but is there a way, or method for ...
1
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0answers
57 views

Multinomial Theorem for Negative Exponents

Using an analog to Newton's binomial theorem with negative exponents, is it true that $$ \begin{align} \left(\sum_{k=0}^mx^k\right)^{-n} & = \sum_{0\le ...
2
votes
2answers
96 views

The digit at the hundred's place of $33^{33}$

I would want to know how to start with the question. And if you get hung up somewhere there's the answer it's $5$. Any help is appreciated thanks, My approach was to look at the factors to somehow ...
0
votes
1answer
12 views

Suppose that number of mistakes on a page is a Poisson RV and independent. From $n$ pages, find the expected number with no mistakes?

A textbook has $n$ pages. The number of mistakes on each page is a Poisson RV with parameter $\lambda$ and is independent of the number of mistakes on all other pages. What is the expected number of ...
2
votes
2answers
31 views

Deriving the variance of a binomial distribution

I know that the variance of a binomial distribution is the number of trials multiplied by the variance of each trial, but I'm not seeing the derivation of this. Here's my logic so far: For each trial ...
0
votes
1answer
36 views

Binomial Distribution with probability $P$ such that $P$ is Uniformly distributed

A number $P$ is random chosen from the uniform distribution from [0,1]. Then a coin with probability $P$ of getting a head is flipped $n$ times. Let $X$ be the number of heads showing and compute ...
0
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1answer
19 views

Finding the $N^{th}$ term In binomial expansion for arbitrary power

How do I find the $n^{th}$ term in the binomial expansion of any index. Ex: Find the $4^{th}$ term of $(1+2x)^{-1/2}$.
1
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1answer
41 views

Find the number of irrational terms in the binomial expansion of $(3^{1/5}+7^{1/3})^{100}$

After expanding the above term binomially, I can well guess that the majority terms are irrational, but i'm unable to find any proper method of solving this sum, after repeated trials. Please help ...
1
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2answers
45 views

Combinatorial Proof of falling factorial and binomial theorem

For $n,m,k \in \mathbb{N}$ is true: $$(n+m)^{\underline{k}}=\sum^{k}_{i=0}\binom ki \cdot n^{\underline{k-i}} \cdot m^{\underline{i}}$$ I can prove the binomial theorem for itself combinatorically ...
-1
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2answers
64 views

Find the coefficient of $x^6$ [closed]

I am practicing the Binomial theorem but now I am stuck over this question. Find the coefficient of $x^6$ in $(1 + x^2 + x^4 + x^6)^{20}$ Please help. Thank you
2
votes
1answer
17 views

Division of the Binomial Coefficient

Prove that when p is prime, the binomial coefficient p!/(r!)((p-r)!) is divisible by p with r being greater than or equal to 1 and less than or equal to p-1 . Clearly p! is divisible by p so I cant ...
0
votes
1answer
23 views

Solving the nth number of this recurrence and cleaning it up using the binomial theorem.

Given this recurrence: an = an-1 – an-2 I was told to create a function that would solve for an. I thus came up with $a_n=\frac{\alpha^{n}-\beta^{n}}{i\sqrt{3}}$ Where ...
1
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1answer
25 views

How many $(r+1)$- subsets of $[n+1]$ have $(k+1)$ as their largest element?

Let $[n+1]$ be the set defined by $[n+1]=\{1,2,\ldots,n+1\}$. Call a subset of $[n+1]$ with $r+1$ distinct elements an $(r+1)$-subset. How many $(r+1)$-subsets of $[n+1]$ have $(k+1)$ as their ...
0
votes
0answers
19 views

Laurent series expansion for powers of n?

I wish to expand the function: $$\dfrac{e^z}{z^n-c^n}$$ about the point $z_0=c$, where c is a constant greater than 0 and n is greater than 2. So I have that $e^{z-c}$ expands to ...
0
votes
1answer
29 views

Find the coefficient of $x^{10}$ in $(x + (\frac{1}{x}))^{100}$

Find the coefficient of $x^{10}$ in $(x + (\frac{1}{x}))^{100}$ My solution: We can calculate the coefficient using the Binomial Theorem: $$(x+y)^n = \sum_{k=0}^n\binom{n}{k}x^{n-k}y^k$$ We know ...
2
votes
1answer
76 views

Compute the sum $\sum_{k\ge0}{k \choose n-k}(-1)^k.$

Compute the sum $$\sum_{k\ge0}{k \choose n-k}(-1)^k.$$ At this point I have tried many binomial theorem identities to try and get something to happen with the $n-k$, but couldn't seem to make ...
1
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4answers
56 views

How find the fractional part of $5^{200}$ divided by $8$?

Finding the fractional part of $\frac{5^{200}}{8}$. I've had this problem given to me (we're learning the Binomial Theorem and all.) So obviously I thought I'd apply the binomial theorem to it, ...
0
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2answers
43 views

On a connection between Newton's binomial theorem and general Leibniz rule using a new method.

In calculus the general Leibniz rule asserts that Let $n$ be a natural numbers, if $f$ and $g$ are $n$-times differentiable functions at a point $x$, then the function $fg$ is also $n$-times ...
1
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3answers
25 views

Calculating the sum of coefficients for binomial expression

The question is: Calculate the sum of the coefficients of $(a-b)^{250}$. My reasoning was that we can take the example of $(a-b)^2$, which would have the coefficients of $1$, $-2$, and $1$, ...
3
votes
1answer
64 views

Dealing with non-constant term in Binomial Theorem question

I am wondering this. Suppose I have a sequence $\{\varepsilon_n\}_{n=0}^\infty$ and elements of this sequence are part of a binomial type expression: For example, my expression is ...
2
votes
2answers
31 views

Compute Using Binomial Theorem [duplicate]

$$\sum_{k=1}^{10} \binom{10}{k} $$ I know the answer is $2^{10} - 1$ but I don't know how to get to the answer.
7
votes
2answers
175 views

What is the ten's digit of $7^{7^{7^{7^7}}}$

What is the ten's digit of $\zeta=7^{7^{7^{7^7}}}$. I got this question while doing binomial theorem. I think that $7^4=2401$ and we only need $\zeta\pmod{100}$. All I could think of is already ...
0
votes
1answer
72 views

Find the last digit of $(\sqrt{71}+1)^{71}+(\sqrt{71}-1)^{71}$

While teaching binomial expansion, one of my high school students asked me the following question: What is the last digit of $(1+\sqrt{71})^{71}+(1-\sqrt{71})^{71}$? I have absolutely no context on ...
3
votes
7answers
382 views

Expansion of complex equation.

Find the value of $$\left(\frac{-1+\sqrt 3i}{2}\right)^{15} + \left(\frac{-1-\sqrt 3i}{2}\right)^{15}.$$ In general, how do we find the value of expansion of equation of high orders other than ...
4
votes
2answers
66 views

Binomial Theorem of Differentiation? [duplicate]

I noticed that $$\frac{d^{n}}{dx^{n}} f(x)g(x)=\sum_{i=0}^n {n \choose i} f^{(i)}(x)g^{(i)}(x)$$ and it's had me scratching for a little bit. It's easy to see how the cross terms add up but can anyone ...
0
votes
1answer
38 views

Binomial theorem, verify ${\frac12 \choose n+1} = 1/(n+1){n-\frac12 \choose n}(-1)^n(1/2)$ [closed]

I need to verify ${\frac12 \choose n+1} = 1/(n+1){n-1/2 \choose n}(-1)^n(1/2)$ and then verify $${2n \choose n}(1/(2^{2n}))={n-1/2 \choose n}$$ Please help!
0
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1answer
29 views

binomial theorem expansion quick question

I'm looking at a proof online to a theorem and some of the steps to the theorem are as follows: let $ x = y+1$ then $$x^p-1 = (y+1)^p -1 = y^p + \sum_{k=1}^{p-1} {p \choose k} y^k$$ $$\iff ...
1
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0answers
34 views

coefficients of polynomial and binomial expressions

Let us say we are given a polynomial p(x)=$\sum_k a_k x^k$. In order to find $\sum_k a_k$ we simply need to evaluate p(1), and similarly there are many other tricks. Is there any trick to evaluate ...
2
votes
2answers
87 views

Find the coefficient of $x^4$ in the expansion of $(1 + 3x + 2x^3)^{12}$?

I have not learnt the multinomial theorem yet, and was trying to approach this using the binomial theorem. I divided the terms as $a$ being $(1+3x)$ and $b$ being $2x^3$. I then used $${12\choose ...
0
votes
2answers
73 views

Determine $(3.12)^5$ to one decimal place.

How many terms in the binomial expansion would be needed to determine $(3.12)^5$ to one decimal place? No calculators allowed by the way. Not really sure how to go about doing it, I could do all the ...
0
votes
1answer
77 views

Proving ${n \choose k}^{-1} = (n+1)\int_0^1 x^k(1-x)^{n-k}\mathsf dx$

Title says it all, I've tried to find the indefinite integral of the right side, got some sort of weird series and got stuck: $$\sum_{i=0}^n-k {n-k \choose i}\cdot{(-1)i\over k+1+i}$$
1
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2answers
29 views

Using the binomial theorem to generate a geometric proof of the derivative.

According to wikipedia, if we wanted to prove $$(x^n)'=nx^{n-1}$$ geometrically by creating an $n$-dimensional hypercube $$(x+\Delta x)^n$$ and setting $a=x$ and $b=\Delta x$, we could expand using ...
0
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0answers
22 views

How to evaluate the sum $\sum_{k=0}^N {{N}\choose {k}} \exp{(\alpha k^2)}$

If $k^2$ in the exponential were replaced with $k$, it is simple to evaluate by the binomial theorem but is it possible to evaluate this sum in closed form with $k^2$?
0
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1answer
19 views

proving $(1+\frac 1n)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem

$(1+\frac 1n)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ this exercise is taken from Apostol's Calculus I (page 45) and it's supposed to be proved by using the binomial ...
0
votes
0answers
31 views

Binomial function [duplicate]

$$S=\sum _{k=1}^n \binom{m+k}{k}$$ As value of n and m can be very big like $10^6 $ or $10^7$, I need S%P where P is a prime. Can anybody help me to derive a closed formula to this sum and also how ...
1
vote
1answer
26 views

Exponents with base close to $1$.

I was just fiddling around with a calculator and calculating powers of numbers really close to $1$ like $1.01,1.001\dots$ trying to find at what value they exceed $2$. This got me thinking if I could ...
0
votes
0answers
16 views

Binomial Expansion without infinity series [Duplicated]

(This is duplicated due do the lack of answers in the previous asking session,because I edited the unreadable math too long that this got lost in "recent questions".I was unable to delete the previous ...
2
votes
1answer
39 views

Binomial Expansion without inifinty series

Variable here is "a" and "b".The question is to simplify the $\sqrt [3] { (\frac{a}{\sqrt{b^3} })\times {\frac {\sqrt{a^6b^2}}{a} }+ \frac{a}{b^2}}$ So these are my steps =$\left(\frac {a^3b} ...