Binomial theorem Definition and 138 Threads

I've started listening to the lectures for the MIT OpenCourseWare 18.01 Single Variable Calculus class. I understood all of it up until the teacher found the derivative of xn. Here's what he wrote on the board: \frac{d}{dx} x^{n} = \frac{\Delta f}{\Delta x} = \frac{(x+\Delta x)^{n} -...

Hey people, I've racked my brain on this question for hours and can't seem to get to grips with it, wondering if i could get a little guidance? Homework Statement Considering the co-efficient of x^n in the expansion of (1+x)^n(1+x)^n, show: [the sum from k=0 -> k=n of:] [nCk]2 = 2nCn...

is it possible to integrate the binomial theorem??

I know how to expand binomials with the aid of pascals triangle and also with the aid of the nCr function on the calculator. I'm not quite sure about this formula though see the part in the brackets where n is above k. What does that mean? Someone told me that represents nCk. Is that true...

Use the Binomial Theorem to show that n summation k-0 3^k C(n,k) = 2^2n hint of the question is 3^k C(n,k) = 3^k 1^n-k C(n,k)

how do i use binomial to show that 3^k C(n,k) = 3^k 1^n-k C(n,k)

Use the binomial expansion of (1+x)^(-1/2) to find an approximation for 1/(rt4.2). I've got the expansion of (1+x)^(-1/2) as 1-(1/2)x+(3/8)x^2... but the obvious idea of substituting x=3.2 gives me the wrong answer. I think it's something to do with the expansion being valid but can't...

Homework Statement Evaluate \sum^{m}_{r=0} ^{ n + r }C_{n} I can handle things when the lower thing in the combination part is changing, what shall I do with this one?

Question: Find the coefficient of x^5 in (1+x+x^2)^4. Problem: I have not come across expanding brackets which have x^2. I know how to apply the binomial theorem for (a+b)^n or (1+a)^n but have not come across (1 + ax + ax^2)^n. They are not explained in my textbooks so I was wondering if...

I am curious, is there any way to use the binomial theorem for fractional exponents? Is there any other way to expand a binomial with a fractional exponent? I suppose Newton's theorem is not a way since it requires factorials. Thanks!

Homework Statement I am trying to find the number of nonnegative integer solutions to a+2b+4c=10^30. I found a generating function, and need to check the coefficient of 10^30. Homework Equations The generating function is 1/((1-x)(1-x^2)(1-x^4)). I found the PFD, which is...

In the binomial expansion (2k+x)^{n}, where k is a constant and n is positive integer, the coefficient of x² is equal to the coefficient of x³ a) Prove that n = 6k + 2 b) Given also that k = .\frac{2}{3}, expand (2k+x)^{n} in ascending powers of x up to and including the term in x³, giving each...

1. Evaluate the numbers for the coefficient of x4y9 in the expansion of (3x + y)13. 2. The Binomial Theorem states that for every positive integer n, (x + y)n = C(n,0)xn + C(n,1)xn-1y + ... + C(n,n-1)xyn-1 + C(n,n)yn. 3. I understand that the coefficients can be found from the n row of...

Prove the following statement: \[ \sum\limits_{r + s = t} {\left( { - 1} \right)^r \left( \begin{array}{c} n + r - 1 \\ r \\ \end{array} \right)} \left( \begin{array}{c} m \\ s \\ \end{array} \right) = \left( \begin{array}{c} m - n \\ t \\ \end{array} \right) \] Any initial...

Use binomial theorem to prove C(n,0) - 3(C(n,1)) + 9(C(n,2) - 27(C(n,3) + ... + (-3)^n(C(n,n) = (-2)^n From looking at the data given b = (-3) so a = 1 so (-2)^n = (1-3)^n With this I know the equation in sigma notation and could probably prove the theorem through mathematical induction but...

I learned the Binomial Theorem a while ago. But it is only now that I think about how it is only useful for powers that are natural numbers. Can it be extended to all real numbers - e.g. 1/2?

Homework Statement I'm having a lot of trouble solving for part b as I am unable to correctly apply the binomial theorem to this approximation. The problem is shown below: Three point charges are distributed: a positive charge +2Q in the center, and a pair of negative charges -Q, a...

Homework Statement Show that if n is a positive integer, then 1\,=\,\binom{n}{0}\,<\,\binom{n}{1}\,<\,\cdots\,<\,\binom{n}{\lfloor\frac{n}{2}\rfloor}\,=\,\binom{n}{\lceil\frac{n}{2}\rceil}\,>\,\cdots\,>\binom{n}{n\,-\,1}\,>\,\,\binom{n}{n}\,=\,1 Homework Equations I think this proof involves...

Ok i need help to calculate the co-efficients of certain terms in the binomial expansion for example: (3 + (5/X)^2)^10 what is the coefficient of x^8? I hope that question works sorry if it doesn't i did just make it up then... if you know of any like it please help! also, an...

There are a few questions that have been giving me trouble with this binomial theorem stuff. (1). Using the binomial theorem and the relation (1+x)^{m_1} (1+x)^{m_2} = (1+x)^{m_1 + m_2} prove that: \sum_{k=0}^n \binom{m_1}{k} \binom{m_2}{n-k} = \binom{m_1 + m_2}{n} (2). Prove by induction...

How would you quickly derive the binomial series? Would you have to use Taylor's Theorem/ Taylor Series? And does the Binomial Theorem follow from the binomial series? Are there any applications at all of the binomial series/ Binomial Theorem to special relativity? I know the binomial series is...

1. For each of the following, simplify so that the variable term is raised is to a single power: (a) State the General Term tr in the Binomial expansion of (2x^2 - 1/x)^10 (b) Find the 7th term in the expansion (c) Is there an x^5 term? Find its coefficient. (d) Is there a constant term...

I'm a complete beginner at these: (3n+1)^3 = (3n)^3 + 3(3n)^2 + 3(3n)^2 + 1 which would give me = 3 (9n^3 + 6n^2) +1 is this correct?

Could anyone tell me what (3n + 2)^x equals to please so I can check my answer? I get something awful that would take me too long to type

I'm having problems figuring out how to do part (b) of this question. a) expand (1-2x)^3 and (1+1/x)^5 b) Find, in the expansion of (1-2x)^3 (1+1/x)^5 i) the constant term ii) the coeffecient of x I've done part a, and I know the formula for a general term for an expansion of a...

anyone could help me with this question... in the expansion of (ax + by)^n, the coeffiients of the first 3 are 6561, 34992, and 81648., Find the value if a, b, and n. i did this... t1 = nC0 (ax)^n = 6561x^n a^n = 6561 t2 = nC1 (ax)^n-1 (by)^1 = 34992x^n-1y bna^n = 34992 but I'm not...

"Show that for small values of x, the function (1+x)^(-1/2) may be approximated by 1-(1/2)x+(3/8)x^2 Hence obtain the approximate value of 1/root(1.01) to 4 decimals." im totally clueless. the example we have isn't well explained at all. can someone even just start me off...

I am trying to do a question from Eugene Hecht's Optics book, which goes something like this: Given the following equations: Cauchy's Equation: n = C_1 + \frac{C_2}{\lambda^2} + \frac{C_3}{\lambda^4} + ... Sellmeier's Equation: n^2 = 1 + \sum_{j}...

Okay, well, perhaps I am already done, perhaps not. This is why I seek your wisdom :) If you are asked to find the first 5 terms of the expansion (1 + 0.07)^9 using the Binomial Theorem, and then asked to use these terms to estimate the value of 1.07^9 what would you put? I added the 5...

Find the term containing x^{20} in (2x - x^4)^{14}. I went t_{k+1}= _{14}C_{k}(2x)^{14-k}(-x^{4})^{k} = 2x^{14-k}(-x^{4k}) First of all, am I on the right track? If so what exactly do I do from there?

Hello Guys, I just have a few quick question on binomial theorem, any help would be greatly appreciated. 1. Expand using the binomial theorm in powers of x up to and including x^3 : (1 + 2x + 3x^2)^5 : I always thought binomial theorm would be used to expand binomials... this is not a...

It seems like this shouldn't be too difficult and yet I'm stumped. I am trying to prove the binomial theorem. (x+y)^n = the sum from k=0 to n of (x^k)*(y^n-k)*(The binomial coefficient n,k) Sorry, about the notation... Anyway, I figure the best way to go about proving this is by...

Hi! I haven't found any good proofs of the binomial theroem. But I've discovered how to go from (a+1)= bla bla to (a+b) = bla bla. So if anyone could told me how to prove (a+1) = bla bla...

The questions are as follows 1) How many terms are there in the expansion of (a+3b-2c-d)^8 before like terms are combined? 4^8 2) How many terms are there after like terms are combined? _8C_4=70 3) What is the coefficent of a^2b^3cd^2? (_8C_2)( _6C_3)( _3C_1)( _2C_2)...

Hi, I'm having some problems with this question - Find the term independent of x in (x^2 - 2/x)^6 I know the answer is something like 6Csomething 2^something, but I'm not sure how to get that. So far I've only really done simple things like (x+y)^n where y is an integer and not...

Binomial Theorem... Hi I need to know about binomial theorem... eg. how to in general expand (a+b)^n I don't understand the combinations / permutations...? thanks Roger

Hey. I am having difficultly with two math problems: 1. In the expansion of (mx+n)^5 the numerical coefficient of the second term is -48 and of the third term if 28.8 Find the values of m and n. 2. The first three terms in the expansion of (1+a)^n are 1-18+144. Determine the values of...

I get a number that is too big when I calculate the coefficient of x^25 in: (2x - (3/x^2))^58 This is how I found the coefficient: Note that (58) = A combination value... k (58) * (2x)^(58-k) * (-3/x^2)^k k (58) * (2)^(58 - k) * (x)^(58 - k) * (-3)^k * (1/x^2)^k k...