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thereddevils
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Homework Statement
Find, in the simplest form, the coefficient of x^n in the binomial expansion of (1-x)^(-6).
Homework Equations
The Attempt at a Solution
i am not sure how to go about with this.
Willian93 said:are u sure you have to find a cofficient contain x^n?
because u should have a specific value for n, so that you could find the cofficient infront of it, or you should at least have which term you are looking for.
HallsofIvy said:Since the exponent, -6, is not a positive integer, you will need to use the generalized binomial series:
[tex](a+ b)^m= \sum_{k=0}^\infty \frac{m(m-1)\cdot\cdot\cdot(m-k+1)}{k!}a^kb^{m-k}[/tex]
Here, of course, a= 1, b= -x, and m= -6 so this is
[tex](1- x)^{-6}= \sum_{k=0}^\infty \frac{-6(-7)\cdot\cdot\cdot(-5-k)}{k!}(-1)^kx^{-6-k}[/tex]
You, apparently, are asked for the coefficient when -6-k= n or when k= -6-n.
The binomial expansion is a mathematical concept that involves expanding a binomial expression raised to a certain power. It is used to find the coefficients of each term in the expanded expression.
The coefficient of x^n in binomial expansion is the number that is multiplied by x^n in the expanded expression. It is represented by the term nCr, where n is the power of x and r is the position of the term in the expansion.
To solve for the coefficient of x^n in binomial expansion, you can use the formula nCr = n! / (r! * (n-r)!), where n is the power of x and r is the position of the term in the expansion. Simply plug in the values and simplify to find the coefficient.
Finding the coefficient of x^n in binomial expansion is important because it helps in simplifying and solving complex mathematical problems. It also allows for easier manipulation of expressions and helps in understanding the patterns and relationships between terms in an expanded expression.
Yes, the coefficient of x^n in binomial expansion can be negative. This can happen when the terms in the expanded expression have alternating signs or when the power of x is an odd number. However, the coefficient is always a whole number, either positive or negative.