Binomial Theorem expansion with algebra

So, (n-2)! cancels out, leaving (n-3) in the denominator.Therefore, we have:6k(n-3)=1Now, we can solve for n:n-3=\frac{1}{6k}n=\frac{1}{6k}+3Since we know that k=\frac{2}{3}, we can substitute that in:n=\frac{1}{6(\frac{2}{3})}+3n=6+3n=9Therefore, we have proven that n=6k+2.
  • #1
thomas49th
655
0
In the binomial expansion [tex](2k+x)^{n}[/tex], 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 [tex]k = .\frac{2}{3}[/tex], expand [tex](2k+x)^{n}[/tex] in ascending powers of x up to and including the term in x³, giving each coefficient as an exact fraction in its simplest form.

my shot at (a)

expand to get x² and x³:

[tex]\stackrel{n}{2}(2k)^{n-2}[/tex] + [tex]\stackrel{n}{3}(2k)^{n-3}[/tex]

subst n = 6k + 2 but i get into more expansion, which i don't think is really going anywhere

can someone help me out/guide me through (a) please?

Thankyou
 
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  • #2
So, do you agree that we must have:
[tex]\frac{n!}{2!(n-2)!}=\frac{n!}{3!(n-3)!2k}[/tex]

Clearly, this can be simplified to:
[tex]\frac{3!2k}{2!}=\frac{(n-2)!}{(n-3)!}[/tex]

Can you simplify this into your desired result?
 
  • #3
i can get it [tex]\frac{3!2k}{2!}=\frac{(n-2)!}{(n-3)!}[/tex] down to 6k(n-3)=1 but that won't simplify to n = 6k + 2.
 
  • #4
Now, you are muddling!
Which number is biggest: (n-2)! or (n-3)!
 
  • #5
(n-2)(n-1) /
(n-3)(n-2)(n-1)

so (n-2)! is cancels

so leaves 1/(n-3)

right?
 
  • #6
No.
We have: (n-2)!=(n-2)*(n-3)!
 

1. What is the Binomial Theorem?

The Binomial Theorem is a mathematical concept that provides a way to expand binomial expressions (expressions with two terms) raised to a certain power. It states that (a + b)^n = a^n + na^(n-1)b + [(n)(n-1)/2!]a^(n-2)b^2 + ... + b^n, where a and b are constants and n is a positive integer.

2. How is the Binomial Theorem used in algebra?

In algebra, the Binomial Theorem is used to expand binomial expressions and simplify them into a polynomial form. This can be helpful when solving equations or simplifying complex expressions.

3. What is the difference between a binomial expression and a polynomial expression?

A binomial expression has only two terms, while a polynomial expression can have more than two terms. The Binomial Theorem is specifically used for expanding binomial expressions, but it can also be applied to polynomial expressions with two terms by setting the remaining terms to zero.

4. Can the Binomial Theorem be used for expressions with variables?

Yes, the Binomial Theorem can be used for expressions with variables. The theorem applies to any expression in the form (a + b)^n, where a and b can be constants, variables, or a combination of both.

5. How do you know when to stop expanding a binomial expression using the Binomial Theorem?

The general formula for the Binomial Theorem includes all possible terms up to the nth power. So, when expanding a binomial expression, you can stop after the nth term, or continue if you need a more precise answer.

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