Problem 35 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Binomial Theorem

The binomial theorem expresses \[(a+b)^n\] as the sum of \[n+1\] terms, each term being in the form of \[\binom{n}{k} a^{n-k} b^k\], where \[k\] ranges from 0 to \[n\]. The 24th term corresponds to \[k = 23\].

Step 2: Determine the Coefficients

The 24th term in \[(a+b)^{25}\] corresponds to \[T_{24} = \binom{25}{23} a^{25-23} b^{23}\]. Calculate the binomial coefficient: \[\binom{25}{23} = \binom{25}{2}\].

Step 3: Compute the Binomial Coefficient

Calculate \[\binom{25}{2}\] using the formula \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]. Thus,\[\binom{25}{2} = \frac{25 \times 24}{2 \times 1} = 300\].

Step 4: Formulate the 24th Term

Substitute the values back into the expression for the term: \[T_{24} = 300a^2b^{23}\].

Key Concepts

Binomial Coefficient

In mathematics, the binomial coefficient is a key part of the binomial theorem. It helps us determine specific terms in the expansion of a binomial expression like \((a+b)^n\). Think of it as a clever way to pick combinations of terms in mathematical form.This concept is denoted as \(\binom{n}{k}\), which stands for the number of ways to choose \(k\) elements from \(n\) elements without considering the order.

• For example, in our expansion of \((a+b)^{25}\), we need the 24th term. Here, the binomial coefficient is \(\binom{25}{23}\).
• Interestingly, \(\binom{25}{23}\) is the same as \(\binom{25}{2}\), due to the symmetric property \(\binom{n}{k} = \binom{n}{n-k}\).

To calculate it, use the formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). In our exercise:

• \(\binom{25}{2} = \frac{25 \times 24}{2 \times 1} = 300\).

This coefficient shows up as a multiplier in front of the specific term in the expansion.

Polynomial Expansion

Polynomial expansion using the binomial theorem involves spreading out a binomial raised to a power into multiple terms. The structure of these terms is predictable, thanks to our reliable binomial coefficients.

• For example, the expression \((a+b)^{25}\) expands into 26 terms, ranging from \(a^{25}\) to \(b^{25}\).
• The general term is \(\binom{n}{k} a^{n-k} b^k\).

This systematic expansion is incredibly useful:

• It simplifies complex calculations into manageable parts.
• Each term in the expansion corresponds to a specific \(k\) value, giving us insight into the structure of the entire polynomial.

In this exercise, finding the 24th term told us to use \(k = 23\) in the formula, where we rearrange it as \(300a^2b^{23}\). This enables us to find any term systematically without expanding everything.

Combinatorics

Combinatorics, a branch of mathematics, provides the tools to count and analyze arrangements of elements. It's the underlying principle behind concepts like binomial coefficients and polynomial expansion.

• The focus is on understanding patterns and combinations.
• It's employed to figure out how many ways events can happen, just like counting the ways to expand a binomial.

Within the context of the binomial theorem:

• Combinatorics helps decide the structure of each term.
• For \((a+b)^{25}\), it helps us determine which power of \(a\) and \(b\) appears in each term.

This clarity in structure translates to real-world applications, where understanding how various components combine can solve complex problems effectively.Combinatorics is thus a powerful mathematical toolkit, ensuring that concepts like binomial expansions are logical and consistent.