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Chapter 11: Problem 65

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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