Problem 65 - Algebra and Trigonometry

Question

Explain how to find a particular term in a binomial expansion without having to write out the entire expansion.

Step-by-Step Solution

Answer

The specific term in a binomial expansion can be calculated using the formula \( ^nC_{r-1} * x^{(n-r+1)} * y^{(r-1)} \). The term number corresponds to the power of y in that term, so use this to find the correct r value. Calculate the expressions and multiply them together.

Step 1: Understanding term position in binomial theorem

First, it's key to know that the term number (r) in the binomial expansion corresponds to the power of y in that term. For instance, in the binomial expansion \( (x+y)^n \), the rth term in the expansion will typically be in the form \( ^nC_{r-1} * x^{(n-r+1)} * y^{(r-1)} \), the r-1 in the formula is due to the fact binomial expansion terms are usually counted starting from 0, not 1. So, for instance, for the 3rd term, place r=2 in the formula.

Step 2: Using binomial theorem to find specific term

Next, use the binomial theorem to calculate this specific term's value. Plug the corresponding r, x and y values of the required term into the formula from step 1 and simplify.

Step 3: Solving for the term

Calculate the values for ^nC_{r-1}, x^{(n-r+1)}, and y^{(r-1)} and then multiply these together to find the term.