Problem 65 - College Algebra

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 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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.