Help Needed with Binomial Expansion Question

The problem asks to show that 3 = (n - 2)k by considering the coefficients of x^2 and x^3. The conversation includes a reminder of the binomial theorem and a suggestion to set the two coefficients equal to solve for n and k. In summary, Cathy is seeking help with a binomial expansion problem involving the coefficients of x^2 and x^3, and is reminded of the binomial theorem and given a suggestion to solve for n and k.
  • #1
CathyLou
173
1
Hi.

I'm completely stuck on the following question, and have no idea how to even start it.

Any help would be really appreciated.

The first four terms, in ascending powers of x, of the binomial expansion of (1 + kx)^n are

1 + Ax + Bx^2 + Bx^3 + ...,

where k is a positive constant and A,B and n are positive intgers.

(a) By considering the coefficients of x^2 and x^3, show that 3 = (n - 2)k.


Thank you.

Cathy
 
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  • #2
Do you know the "binomial theorem":
[tex](a+ b)^n= \sum_{i=0}^n _nC_i a^i b^{n-i}[/tex]
where [itex]_nC_i[/itex] is the "binomial coefficient" n!/i!(n-i)!.

In particular, the coefficient if xi in (1+ kx)^n is [itex]_nC_i k^i[/itex]
Here, you are GIVEN that the coefficient of x2, which is k2n(n-1)/2, and the coefficient of x3, which is k3n(n-1)(n-2)/6 are equal. Set them equal and cancel everything you can.
 
  • #3
Okay. Thanks very much for your help!

Cathy
 

1. What is binomial expansion?

Binomial expansion is a mathematical technique used to expand a binomial expression, which is an algebraic expression with two terms, raised to a certain power. It involves using the binomial theorem to find the coefficients of each term in the expanded expression.

2. How do I solve a binomial expansion question?

To solve a binomial expansion question, you can use the binomial theorem or the Pascal's triangle method. The binomial theorem involves using a formula to find the coefficients of each term, while the Pascal's triangle method involves using a triangular array to find the coefficients.

3. What is the binomial theorem?

The binomial theorem is a mathematical formula that is used to expand a binomial expression raised to any power. It states that the coefficients of each term in the expansion can be found by using combinations and powers of the two terms in the binomial expression.

4. How is binomial expansion used in real life?

Binomial expansion is used in various fields such as statistics, economics, and engineering. In statistics, it is used to calculate probabilities and in economics, it is used to model growth and decay. In engineering, it is used to analyze and design circuits and systems.

5. What are some common mistakes to avoid when solving binomial expansion questions?

Some common mistakes to avoid when solving binomial expansion questions include forgetting to distribute the exponent to each term, using incorrect formula or method, and not simplifying the final expression. It is important to carefully follow the steps and check for errors in calculations to ensure accurate solutions.

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