Problem 34 - Precalculus: Mathematics for Calculus

Question

Find the indicated terms in the expansion of the given binomial. The fifth term in the expansion of \((a b-1)^{20}\).

Step-by-Step Solution

Answer

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

Step 1: Identify the General Term Formula

In a binomial expansion, the general term can be given by the formula: \[ T_{k+1} = \binom{n}{k} (a^b)^{n-k} (-1)^k \]where \(a^b\) represents the first term in the binomial, \(-1\) is the second term, \(n\) is the total number of terms, and \(k\) is the term number starting from zero. In this case, \(n = 20\).

Step 2: Set Up for the Fifth Term

To find the fifth term, we set \(k = 4\) since terms are counted starting from zero. Plug \(k = 4\) into the general term formula:\[ T_{5} = \binom{20}{4} (a b)^{20-4} (-1)^4 \]

Step 3: Calculate the Binomial Coefficient

Compute the binomial coefficient \(\binom{20}{4}\):\[ \binom{20}{4} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845 \]

Step 4: Evaluate the Power of Terms

Calculate the powers involved:- \((a b)^{16} = a^{16}b^{16}\)- \((-1)^4 = 1\)

Step 5: Write the Fifth Term

Combine the results from the previous steps:\[ T_{5} = 4845 \times a^{16} b^{16} \times 1 = 4845 a^{16} b^{16} \]

Key Concepts

Understanding Binomial Coefficient

The binomial coefficient is a crucial part of binomial expansion. It is defined as the number of ways to choose a subset of elements from a larger set, without regard to the order of elements in the subset. Mathematically, it is represented as \( \binom{n}{k} \), which is read as "\( n \) choose \( k \)". This formula calculates the coefficients in the expansion of a binomial expression.

• \( n \) is the total number of items or the power of the binomial.
• \( k \) stands for the specific term's position (starting from a zero index).
• The binomial coefficient is calculated as:

\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Here, the exclamation mark \(!\) denotes factorial, which is the product of all positive integers up to that number. In the context of the original exercise, the fifth term of \((ab - 1)^{20}\) requires calculating the value \( \binom{20}{4} \), which equals 4845. This coefficient tells us how many times the specific product will appear in the expansion.

Deriving the General Term Formula

The general term formula is a versatile tool used to find any term of a binomial expansion without expanding the entire expression. It is useful especially for large powers. The binomial expansion formula for the \( k+1 \)th term is:

• \( T_{k+1} = \binom{n}{k} (a^b)^{n-k} (-1)^k \)
• \( a^b \) is the coefficient of the first term in the binomial.
• \((-1)^k\) results from the second term \(-1\) in the given expression \((a b - 1)^n\).

This formula breaks down each term into its constituents, consisting of coefficients determined by binomial coefficients, and powers based on the index of the term sought. For the given problem, to find the fifth term, we substitute \( k = 4 \) in the formula, giving us the term with the coefficients, and powers of \( a \), \( b \), and \( -1 \). This structured approach eases computation.

Calculating Power of Terms

In a binomial expression like \(( ab - 1)^n\), the "power of terms" refers to the exponent to which each individual component of the binomial is raised. Each term in a binomial expansion involves raising the two elements of the binomial to various powers, which were predetermined by their position in the sequence of terms:

• The power of \( a^b \) decreases as \( k \) increases.
• Conversely, the power of \( -1 \) increases with \( k \).

For step 4 in the solution, we find the powers for the term index: \( (a b)^{16} = a^{16} b^{16} \) and \((-1)^4 = 1\). Here, balancing powers of \( a \) and \( b \) with negative terms ensures accurate representation of each term.

Exploring the Binomial Theorem

The binomial theorem is a formula that provides a quick way to expand a binomial raised to any power. It states that\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]This theorem provides every possible combination of the powers of \( a \) and \( b \), combined with their respective binomial coefficients.

• The expression is expanded as a sum of terms.
• Each term is a product involving powers of \( a \) and \( b \).
• Terms are controlled by \( k \), the term index, which varies from 0 to \( n \).

Using the binomial theorem allows us to find any term, such as the fifth term in an expansion, without listing all previous terms. For students, understanding this theorem is essential because it lays the foundation for deeper studies in algebra and calculus.