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Chapter 8: 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 required term in the binomial expansion can be found by understanding and applying the formula of the general term of a binomial expansion, \( T_(r+1) = ^nC_r \cdot a^(n-r) \cdot b^r \). Substitute the values of 'r', 'a', 'b', and 'n' into the formula, perform the necessary calculations, and simplify the solution.

Step by step solution

01

Understand the General Term of a Binomial Expansion

The general term of a binomial expansion is given by the formula \( T_(r+1) = ^nC_r \cdot a^(n-r) \cdot b^r \) where \( T_(r+1) \) is the (r+1)th term in the expansion of \( (a+b)^n \), ^nC_r represents n choose r (combination), a is the first term in the binomial, b is the second term, and n is the power the binomial is raised to.
02

Identifying the Required Term

Identify the term number that needs to be found. Remember, if you have to find the kth term, you should use k-1 in place of r in the formula. This is due to the term number being one more than the index.
03

Applying the Formula

Substitute the identified term number, base 'a' and 'b', and power 'n' into the formula. Calculate ^nC_r, a^(n-r) and b^r separately. Multiply these three results together to obtain the required term.
04

Simplifying the Solution

After calculations, simplify the solution if possible. The final term obtained is the required term in the binomial expansion without fully expanding it.

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