Problem 35 - 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: Understand the Binomial Theorem

The expansion of a binomial expression \((a+b)^n\) is given by the Binomial Theorem, which states that the r-th term in the expansion is given by \(T_{r+1} = \binom{n}{r} a^{n-r} b^r\). Here, \(n\) is the power of the binomial, \(a\) and \(b\) are the terms of the binomial, and \(r\) is the term number minus one.

Step 2: Identify Given Values

We are given \((a+b)^{25}\) and we need to find the 24th term. This means we will use \(n=25\) and the term number \(r+1 = 24\), so \(r = 23\).

Step 3: Apply the Binomial Formula

Use the formula \(T_{r+1} = \binom{n}{r} a^{n-r} b^r\). Plugging in the values, \(n = 25\) and \(r = 23\), we get: \[ T_{24} = \binom{25}{23} a^{25-23} b^{23} = \binom{25}{23} a^2 b^{23} \].

Step 4: Calculate the Binomial Coefficient

The binomial coefficient \(\binom{25}{23}\) is equivalent to \(\binom{25}{2}\) due to symmetry, and it calculates as:\[ \binom{25}{2} = \frac{25 \times 24}{2} = 300 \].

Step 5: Putting It All Together

Substitute back into the formula from Step 3: \[ T_{24} = 300 a^2 b^{23} \]. This is the 24th term in the expansion of \((a+b)^{25}\).

Key Concepts

Understanding Binomial Expansion

Binomial expansion is an essential concept in algebra that allows us to express powers of a binomial in expanded form. This means when you have an expression like \((a+b)^n\), you can expand it to show all its terms. Each term in the expansion has specific coefficients called binomial coefficients, which we'll talk about next. Binomial expansion is widely used in algebra to simplify expressions and solve equations.

• The expanded form shows each term separated, often used to evaluate special cases such as specific terms in a sequence.
• The number of terms in the expansion is always one more than the power, so \((a+b)^n\) has \((n+1)\) terms.
• This concept also helps students understand symmetry and combinations, important aspects in mathematical studies.

By mastering binomial expansion, you can efficiently break down more complex polynomial expressions into understandable parts.

Exploring Binomial Coefficients

Binomial coefficients are the numbers that appear in the expanded form of a binomial expression. In the expression \((a+b)^n\), each term is a multiple of a binomial coefficient denoted as \(\binom{n}{r}\). These coefficients are calculated using combinations, representing the number of ways to choose \(r\) elements from a set of \(n\) elements.

• For a term \(T_{r+1}\) in the expansion, the binomial coefficient is calculated as \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(!\) denotes a factorial.
• They are symmetrical, meaning \(\binom{n}{r} = \binom{n}{n-r}\).
• This symmetry within binomial coefficients plays a significant role in simplifying complex calculations in combinatorics and algebra.

Understanding these coefficients helps resolve challenging algebraic problems, particularly in calculating specific terms in a sequence.

Algebraic Expressions in Binomial Expansions

Algebraic expressions are combinations of constants, variables, and mathematical operations like addition or multiplication. In binomial expansions, we often deal with two variables, such as \((a+b)\) raised to a power. It's crucial to understand how these elements interact.

• Each term in a binomial expansion can be seen as a distinct algebraic expression, consisting of variables raised to specific powers multiplied by a binomial coefficient.
• The interaction of variables and numbers follows a pattern determined by the laws of exponents and factorization.
• Recognizing the pattern in which algebraic expressions unfold in binomial expansion enables simplification and evaluation of terms more accurately.

Having a solid grasp of algebraic expressions is vital when dealing with binomial expansions, as these concepts often appear together in various mathematical problems.