Problem 30 - Precalculus: Mathematical for Calculus

Question

Find the fifth term in the expansion of \((a b-1)^{20}\)

Step-by-Step Solution

Answer

The fifth term is \(4845 (ab)^{16}\).

Step 1: Understand the Context

The problem requires finding a specific term in the binomial expansion of \((ab-1)^{20}\). We will use the Binomial Theorem for this purpose.

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \((x+y)^n = \sum_{k=0}^{n} {\binom{n}{k} x^{n-k} y^k}\). Here, substituting \(x = ab\), \(y = -1\), and \(n = 20\), we can express each term of the expansion as: \(T_k = \binom{20}{k} (ab)^{20-k} (-1)^k\).

Step 3: Determine the Required Term Index

The formula tells us that the \(k\)-th term is \(\binom{20}{k-1} (ab)^{21-k} (-1)^{k-1}\). To find the 5th term, set \(k-1 = 4\), so \(k = 5\). Therefore, we find the term with \(k=5\).

Step 4: Calculate the 5th Term

For \(k = 5\), the term is given by: \[T_5 = \binom{20}{4} (ab)^{16} (-1)^4\]First, calculate the binomial coefficient: \(\binom{20}{4} = \frac{20 \cdot 19 \cdot 18 \cdot 17}{4 \cdot 3 \cdot 2 \cdot 1} = 4845\). Since \((-1)^4 = 1\), the 5th term becomes: \[T_5 = 4845 (ab)^{16}\].

Step 5: Conclude the Solution

The 5th term in the expansion of \((ab-1)^{20}\) is \(4845 (ab)^{16}\).

Key Concepts

Understanding Binomial Coefficient

A key component in solving our problem is the binomial coefficient. It appears when we expand expressions using the Binomial Theorem. The binomial coefficient is denoted as \( \binom{n}{k} \), and it represents the number of ways to choose \( k \) elements from a set of \( n \) elements.
To compute \( \binom{20}{4} \), which we needed for the problem, you use the formula:

• \( \binom{n}{k} = \frac{n!}{k! (n-k)!} \)

This formula calculates the total combinations possible, considering how order doesn't matter in combinations. In our exercise, plugging in the values gives us:

• \( \binom{20}{4} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845 \)

When solving similar problems, spotting these coefficients can help you immediately identify important factors in the expansion.

Term Expansion in Binomial Theorem

When dealing with a binomial expansion like \((ab-1)^{20}\), understanding how each term is expanded is crucial. The Binomial Theorem provides a way to express such an expansion through a series of terms.
Each term in the expansion is given by:

• \( T_k = \binom{n}{k} x^{n-k} y^k \)

In our specific problem, substituting \(x = ab\), \(y = -1\), and \(n = 20\) helps to derive each term. With this substitution:

• \( T_k = \binom{20}{k} (ab)^{20-k} (-1)^k \)

This method systematically allows you to find any term in the expansion without fully expanding the whole expression.

Powers of Expressions

In the context of binomial expansions, understanding how powers of expressions behave is essential. When expressing each term, you raise parts of the initial binomial, like \((ab)\) and \((-1)\), to certain powers based on the specific term.
For example, in our problem to find the fifth term, we calculate \((ab)^{16}\). This means multiplying \(ab\) by itself 16 times. Similarly, any negative part of the term, \((-1)^k\), needs to be evaluated:

• Negative powers will flip the sign for odd \(k\)
• Positive powers will retain the positive sign for even \(k\)

In our case, \((-1)^4 = 1\), showing us that multiplying negatives an even number of times results in a positive value. Understanding these principles ensures accurate calculations in the expansion.