Problem 29 - Precalculus: Mathematical 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 \(48620x^{18}\).

Step 1: Determine Total Number of Terms

The total number of terms in the expansion of a binomial expression \((a+b)^n\) is given by \(n+1\). Here, \(n=18\) for the expression \((x^2+1)^{18}\). Thus, the number of terms is \(18+1=19\).

Step 2: Identify the Middle Term Position

Since the total number of terms is 19, an odd number, the middle term is the 10th term. This is calculated by taking \((19 + 1) / 2 = 10\).

Step 3: Formulate the General Term of the Expansion

The general term \(T_k\) in the expansion of \((a+b)^n\) is given by: \[ T_k = \binom{n}{k-1}a^{n-(k-1)}b^{k-1} \] In this case, \(a = x^2\), \(b = 1\), and \(n=18\).

Step 4: Substitute Values to Find the Middle Term

Substitute \(n=18\), \(a=x^2\), \(b=1\), and \(k=10\) into the formula for the general term: \[ T_{10} = \binom{18}{9}(x^2)^{18-9}(1)^9 \] This simplifies to: \[ T_{10} = \binom{18}{9}(x^2)^9 \]

Step 5: Simplify the Middle Term Expression

Calculate the coefficient \(\binom{18}{9}\) and simplify the power of \(x\): \[ \binom{18}{9} = 48620 \] and \((x^2)^9 = x^{18}\). Thus, the middle term is: \[ T_{10} = 48620 \cdot x^{18} \]

Step 6: State the Middle Term

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

Key Concepts

Binomial Theorem

The Binomial Theorem is a foundational concept in algebra that provides a formula for expanding expressions of the form \((a+b)^n\). Essentially, it offers a way to expand any binomial expression without needing to multiply everything out step by step. The theorem states that:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• Where \(\binom{n}{k}\) denotes the binomial coefficient, which is calculated as the number of ways to choose \(k\) elements from \(n\) elements, giving us \(\frac{n!}{k!(n-k)!}\).

This allows us to determine any specific term in the expansion by inserting values into the formula. For example, if you know the specific term you want, you can plug it in as \(T_k = \binom{n}{k-1}a^{n-(k-1)}b^{k-1}\).
The Binomial Theorem is particularly helpful for large powers, where manual expansion would be cumbersome.

Middle Term

Finding the middle term in a binomial expansion can be useful for simplifying problems and gaining insights into the properties of the polynomial.
In any expansion, knowing the number of terms is vital. For \((x^2 + 1)^{18}\), the number of terms can be calculated as \(n+1\), which comes to 19 terms in this case.
When the number of terms is odd, the middle term is easily determined by the formula \(\text{Middle Term Position} = \frac{n+1}{2}\). So, for 19 terms, the middle term is the 10th term. Once identified, the middle term can drastically simplify the problem, such as calculating the power of a specific term without expanding the entire expression.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, permutations, and counting principles. The Binomial Theorem heavily relies on combinatorial concepts, especially when determining binomial coefficients.
The binomial coefficient \(\binom{n}{k}\) is a basic combinatorial tool. It calculates how many ways you can choose \(k\) items from a larger set of \(n\) items, regardless of order. The formula is:

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

This formula is not just theoretical; it is applied in practical problems where arrangements or groupings are relevant.
Combinatorial principles appear throughout various areas of mathematics and are essential for simplifying and solving problems, whether finding terms in a binomial expansion or solving real-world logistical questions.