Problem 27 - Algebra and Trigonometry

Question

Find the last two terms in the expansion of \(\left(a^{2 / 3}+a^{1 / 3}\right)^{25}\)

Step-by-Step Solution

Answer

The last two terms are \(25a^{24+\frac{2}{3}}\) and \(a^{\frac{25}{3}}\).

Step 1: Understand the Binomial Theorem

The binomial theorem states that \((x + y)^n\) can be expanded as \(\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). In this problem, \(x = a^{2/3}\) and \(y = a^{1/3}\) with \(n = 25\).

Step 2: Identify the Expression for General Term

The general term in the expansion \(T_k\) is given by \(\binom{25}{k} (a^{2/3})^{25-k} (a^{1/3})^k\). Simplifying the powers of \(a\), we have \(T_k = \binom{25}{k} a^{\frac{50-k}{3}} a^{\frac{k}{3}}\), which simplifies to \(T_k = \binom{25}{k} a^{\frac{50}{3}}\).

Step 3: Simplify Powers and Combine Terms

Combine the powers of \(a\): \(a^{50/3-k/3+k/3} = a^{(50+k)/3}\). So, the term becomes \(T_k = \binom{25}{k} a^{(50+k)/3}\).

Step 4: Find the Second-to-Last Term

We need to find terms for \(k = 24\) and \(k = 25\) as they will yield the last two terms in the expansion. For \(k = 24\), the term is \(\binom{25}{24} a^{\frac{74}{3}}\) or simply, \(\binom{25}{24} a^{24 + \frac{2}{3}}\).

Step 5: Find the Last Term

For \(k = 25\), the term becomes \(\binom{25}{25} a^{25/3} = \binom{25}{25} a^{\frac{25}{3}}\). Since \(\binom{25}{25} = 1\), the term is \(a^{\frac{25}{3}}\).

Key Concepts

Binomial Expansion

Binomial expansion is a powerful tool in algebra that helps us expand expressions raised to a power. The binomial theorem gives us a formula for expanding expressions of the form \((x + y)^n\). The formula is:\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]In simpler terms, it tells us how to distribute the powers to the terms inside the parentheses, step by step. In our example, - We have \(x = a^{2/3}\) and \(y = a^{1/3}\) raised to the power of 25.- This expression can be expanded using the binomial formula by plugging these values into the expression.The binomial coefficient \(\binom{n}{k}\) is used to determine the number of ways to choose \(k\) items from \(n\) items without regard to order. This concept is pivotal for generating the various terms in the expansion.

General Term Identification

Identifying the general term in a binomial expansion is the key to understanding each specific component of the expanded form. The general term in our case can be represented as:\[T_k = \binom{25}{k} (a^{2/3})^{25-k} (a^{1/3})^k\]The index \(k\) determines which term we are calculating, while the binomial coefficients \(\binom{25}{k}\) provide numerical coefficients for each term. In simplifying the general term, combine the powers of \(a\). Simplifying the expression for the powers, we get:\[T_k = \binom{25}{k} a^{\frac{50-k}{3}} a^{\frac{k}{3}}\]This results in further simplification:\[T_k = \binom{25}{k} a^{(50+k)/3}\]

Combining Powers

Combining powers in binomial expansions is crucial to simplify each term in the sequence. Once we understand the exponents in our general term, we need to add them together sensibly. Given our expression:\[a^{\frac{50-k}{3}} a^{\frac{k}{3}}\]Combining these, we have:\[a^{(50-k)/3 + k/3}\]The simplified form is \[a^{(50+k)/3}\]. This step ensures that each term's power of \(a\) is consolidated for easy calculation. Through this method, each term in the expansion is easily identified and can be evaluated numerically if needed.

Last Terms in Expansion

Finding the last terms in a binomial expansion is often the critical task, especially when we're interested in endpoints or extremes. In our particular problem, we need to find the last two terms for \(k = 24\) and \(k = 25\). - **For \(k = 24\)**: The term is: - \[T_{24} = \binom{25}{24} a^{\frac{74}{3}}\]- **For \(k = 25\)**: The term is: - \[T_{25} = \binom{25}{25} a^{\frac{25}{3}} = a^{\frac{25}{3}}\]To understand why these are the last terms:- The largest power of \(a\) in the expansion decreases as \(k\) increases. - When \(k = 25\), the power of \(a\) is smallest, marking it as the last term.These insights facilitate a deeper grasp of the expansion’s structure.