Problem 31 - Algebra and Trigonometry

Question

Find the 24 th term in the expansion of \((a+b)^{25}\)

Step-by-Step Solution

Answer

The 24th term is \(300 a^2 b^{23}\).

Step 1: Identifying the Formula

We need to use the Binomial Theorem to find the specific term in the expansion of \((a+b)^{25}\). The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).

Step 2: Determine the Term Number

We need the 24th term of the expansion. The term number in the Binomial expansion is given by \(k+1\). Therefore, for the 24th term, \(k = 23\).

Step 3: Substitute Values into the Formula

Using \(k = 23\) and \(n = 25\) from the expansion formula, the 24th term is \(\binom{25}{23} a^{25-23} b^{23}\).

Step 4: Calculate the Binomial Coefficient

The binomial coefficient \(\binom{25}{23}\) is calculated using \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Thus, \(\binom{25}{23} = \frac{25!}{23!\cdot2!} = 300\).

Step 5: Express the Term

Substitute the values back into the expression: the 24th term is \(300 a^{2} b^{23}\).

Key Concepts

Binomial Coefficient

The Binomial Coefficient is a crucial part of the Binomial Theorem and is used to determine the coefficients of terms in a binomial expansion. It is denoted as \( \binom{n}{k} \) and pronounced as "n choose k." This coefficient tells us how many ways we can choose \(k\) elements from a set of \(n\) elements. It plays a vital role in combinatorics, which is a branch of mathematics dealing with combinations and arrangements.

The formula to calculate the Binomial Coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \(!\) represents factorial, which means the product of all positive integers up to that number. For example, 5! is 5 \( \times \) 4 \( \times \) 3 \( \times \) 2 \( \times \) 1. Understanding how to compute factorials is important for calculating the Binomial Coefficient correctly. In our exercise, we calculate \( \binom{25}{23} \) using the formula to find it equals 300.

• The Binomial Coefficient tells us how many ways we can arrange particular elements.
• It provides the coefficients for terms in binomial expansions.
• Understanding factorials is key to calculating it.

Polynomial Expansion

Polynomial Expansion is the process of expressing a binomial expressed as a power, such as \((a+b)^n\), as a sum of terms. Each term is constructed by multiplying binomial coefficients by the power of each variable. The expansion follows a predictable pattern dictated by the Binomial Theorem. Understanding this process is essential in algebra, especially in solving equations that involve powers.

In the general form, \((a + b)^n\) expands to \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). This tells us:

• The number of terms in the expansion is \(n+1\).
• Each term consists of a binomial coefficient, a power of \(a\), and a power of \(b\).
• The powers of \(a\) decrease while the powers of \(b\) increase as you move through each term.

In our example, we expand \((a+b)^{25}\), and use the formula to specifically target the 24th term, showing how powerful and systematic polynomial expansions can be.

Algebraic Expressions

Algebraic Expressions are combinations of variables, numbers, and operators (such as plus signs). They can represent real-world quantities or be part of mathematical operations. These expressions form the foundation of algebra and are crucial for creating equations from various scenarios.

When dealing with algebraic expressions in binomial expansions, understanding how to manipulate and simplify expressions is essential. In our exercise, the expression \(300 a^2 b^{23}\) represents the 24th term of the binomial expansion. Knowing how to work with expressions allows us to decipher the structure and value of each term in an expansion.

• Algebraic expressions consist of terms with variables and coefficients.
• Understanding these basics helps in manipulating polynomials and solving equations.
• They are the building blocks of more complex algebraic concepts like polynomial expansions.

Managing and simplifying such expressions is an invaluable skill, especially as math problems become more complex.