Problem 33 - Precalculus Mathematics for Calculus

Question

Find the middle term in the expansion of \(\left(x^{2}+1\right)^{18}\).

Step-by-Step Solution

Answer

The middle term is \(48620 x^{18}\).

Step 1: Recognize the Binomial Expansion Structure

The expansion of \(\left(x^2 + 1\right)^{18}\) follows the binomial theorem: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]where \(a = x^2\), \(b = 1\), and \(n = 18\).

Step 2: Identify the Middle Term in a Binomial Expansion

For a binomial expansion of order \(n\), the middle term is the \(\frac{n}{2} + 1\)-th term when \(n\) is even. Since \(n = 18\), an even number, the middle term is the \(\frac{18}{2} + 1 = 10\)-th term.

Step 3: Determine the General Term

The general term \(T_k\) in a binomial expansion is given by:\[T_k = \binom{n}{k} a^{n-k} b^k\]For our expansion, \(T_{10} = \binom{18}{9} (x^2)^{18-9} (1)^9 = \binom{18}{9} (x^2)^9 \).

Step 4: Simplify the Middle Term

Simplify \(T_{10}\):\[T_{10} = \binom{18}{9} (x^2)^9 = \binom{18}{9} x^{18}\]Utilize the binomial coefficient: \[\binom{18}{9} = 48620\]}, as calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), giving \(T_{10} = 48620 x^{18}\).

Step 5: State the Middle Term

The middle term of the binomial expansion of \(\left(x^2 + 1\right)^{18}\) is \(48620 x^{18}\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that provides a way to expand expressions of the form \((a + b)^n\). This theorem is essential for calculating each term in the expansion without multiplying all the terms manually. It is expressed in the formula:

• \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]

This formula tells us that each term combines powers of \(a\) and \(b\) determined by \(n\) and \(k\). Each term in the expansion is also multiplied by a specific number called the binomial coefficient. It is determined by the pair \((n, k)\), making it an important part of expanding any binomial expression. It systematically lays out the emergence of terms by increasing the exponent of \(b\) and decreasing the exponent of \(a\).
Understanding this theorem not only simplifies the expansion of polynomials but also introduces students to the concept of combinations, a fundamental principle in combinatorics, which counts the number of ways to choose items.

Middle Term Binomial Expansion

In a binomial expansion, finding the middle term can save time and effort, especially for larger powers such as in \((x^2 + 1)^{18}\). The method to identify this middle term depends on whether the power \(n\) is even or odd. When \(n\) is even, like our problem where \(n = 18\), the middle term is the \(\frac{n}{2} + 1\)-th term.
Here, dividing 18 by 2 gives 9. Adding 1 to this result gives 10, identifying the middle term as the 10th term in the series. This approach is helpful because it lets you pinpoint the central term without expanding all possible terms entirely, providing efficiency and a strategic way to approach binomial expressions.

Binomial Coefficient

A crucial component of each term in a binomial expansion is the binomial coefficient. Denoted as \(\binom{n}{k}\), it represents the number of ways to select \(k\) elements from \(n\) total elements, which is calculated using the formula:

• \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

For the middle term in our example, we computed \(\binom{18}{9}\), calculated as:

• \[\frac{18!}{9!9!}\]

This came out to be 48620, indicating not just how the terms increase or decrease but their corresponding weights.
Binomial coefficients are fundamental not only in algebra but also in probability and statistics, as they link directly to the concept of combinations and are related to Pascal's Triangle, where each level corresponds to the coefficients of a binomial expansion.

Mathematical Proofs

Mathematical proofs are logical arguments that verify the truth of mathematical statements. In the context of binomial expansions, they ensure that each derived term and coefficient is accurate. Proofs in this field often involve demonstrating properties of combinations and factorials.
For instance, proving the identity of binomial coefficients using properties of \(n!\), involves showing that \(\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}\), and using the relation between different coefficients to prove identities in algebra.
Strong proofs help mathematicians solve more complex problems and solidify understanding, a necessary skill for progressing deeper into mathematical analysis and related topics, showing that coherent steps lead to valid conclusions based on initial premises.