Problem 33 - Algebra and Trigonometry

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: Understand the Binomial Theorem

The Binomial Theorem provides a way to expand expressions of the form \((a + b)^n\). It states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this problem, \(a = x^2\), \(b = 1\), and \(n = 18\).

Step 2: Determine the Total Number of Terms

The number of terms in the binomial expansion of \((x^2 + 1)^{18}\) is \(n + 1 = 18 + 1 = 19\). Since 19 terms are odd, the middle term is the 10th term.

Step 3: Use the Binomial Theorem to Find the Middle Term

The middle term (10th term) corresponds to \(k=9\) in the binomial expansion. According to the Binomial Theorem, the general term is given by \(T_k = \binom{18}{k} (x^2)^{18-k} (1)^k\). For the 10th term: set \(k=9\).

Step 4: Calculate the Middle Term

Substitute \(k=9\) into the general term formula: \[T_{10} = \binom{18}{9} (x^2)^{18-9} (1)^9 = \binom{18}{9} (x^2)^9\]Calculate each part: - \(\binom{18}{9} = 48620\) (using a binomial coefficient calculator).- \((x^2)^9 = x^{18}\).Therefore, \(T_{10} = 48620x^{18}\).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful method used to expand expressions of the form \((a + b)^n\). This expansion is articulated through the Binomial Theorem. The theorem states: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, each term of the expansion can be determined as a combination of the terms \(a\) and \(b\).
For example, take the problem of expanding \((x^2 + 1)^{18}\) as presented above.

• In this expansion, \(a = x^2\) and \(b = 1\)
• The total number of terms will be \(n + 1\), which equals 19 in this case.

Each term can be represented in the form of \( \binom{n}{k} a^{n-k} b^k \), where \( \binom{n}{k} \) is a binomial coefficient.
By understanding this formula, you can determine any specific term in the expansion by choosing the appropriate value of \(k\).

Middle Term

When dealing with a binomial expansion, the middle term is the term that sits exactly at the center of the sequence of terms.
In a binomial expansion with an odd number of terms, the middle term is the one that is equally distant from both the start and the end of the term sequence.
For the binomial expansion of \((x^2 + 1)^{18}\), there are 19 terms.

• The middle term is thus the 10th term.

Finding the middle term involves setting \(k\) such that it corresponds to this central position. As seen in the problem, the 10th term corresponds to \(k=9\).
Substituting \(k=9\) into the general formula yields the middle term from the expansion.

Binomial Coefficients

Binomial coefficients are crucial in the expansion using the Binomial Theorem. They represent the number of ways to choose \(k\) elements from a set of \(n\) elements.
The coefficients are denoted as \( \binom{n}{k} \) and are determined using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
In simpler terms, this formula gives us the value of the binomial coefficient for a particular term in the expansion.
For the middle term of the binomial \((x^2 + 1)^{18}\), where \(k=9\), we calculate: \[ \binom{18}{9} = \frac{18!}{9!\times9!} = 48620 \]

• This is the coefficient of the term \((x^2)^9\).
• These coefficients not only determine the weight of each term but also indicate how these terms contribute to the complete expansion.

Understanding binomial coefficients can enhance your comprehension of each step within the Binomial Theorem.