Binomial Theorem: Set 2

Learning Objectives Find the general term (the (r+1)th term) in the expansion of (ax + by)^n. Calculate a specific term (e.g., the 5th term) in a binomial expansion. Determine the coefficient of a specific power of a variable (e.g., the coefficient of x^7). Find the term independent of a variable in a binomial expansion. Identify and calculate the middle term(s) in an expansion for both even and odd values of n. Apply the Binomial Theorem to expansions involving fractions and negative terms. How could you find the 10th term in the expansion of (x + 2y)^25 without writing out all 26 terms? 🤔 In 'Set 1', you learned to expand a full binomial. Now, we'll learn powerful shortcuts to find specific pieces of an expansion, like a single term or a specific coefficie...

TermDefinitionExample General Term (T_{r+1})A formula that represents any term in a binomial expansion based on its position. It allows us to calculate a specific term without computing the entire expansion.In the expansion of (a+b)^n, the general term is T_{r+1} = \binom{n}{r} a^{n-r} b^r. To find the 3rd term, we use r=2. Binomial CoefficientThe numerical part of each term in a binomial expansion, represented by \binom{n}{r} or nCr. It is calculated as \frac{n!}{r!(n-r)!}.In (x+y)^4, the coefficient of the term with y^2 is \binom{4}{2} = \frac{4!}{2!2!} = 6. Index (n)The power to which the binomial is raised. It determines the number of terms in the expansion (n+1).For (2x - 5)^7, the index n is 7. The expansion will have 7+1 = 8 terms. Term Independent of a VariableThe constant term in...

The General Term Formula For the expansion of (a+b)^n, the (r+1)th term is: T_{r+1} = \binom{n}{r} a^{n-r} b^r This is the most important formula for this lesson. Use it to find any specific term. Remember that for the k-th term, you must set r = k-1. Finding the Middle Term (n is even) The middle term is the (\frac{n}{2} + 1)^{th} term. When the index 'n' is an even number, there is only one middle term. Use this formula to find its position, then use the General Term Formula to calculate it. Finding the Middle Terms (n is odd) The two middle terms are the (\frac{n+1}{2})^{th} and (\frac{n+3}{2})^{th} terms. When the index 'n' is an odd number, there are two middle terms. Use these formulas to find their positions, then calculate each one.