Problem 10 - Discrete Mathematics with Applications

Question

Find the middle term in the binomial expansion of each. $$\left(x-\frac{1}{x}\right)^{6}$$

Step-by-Step Solution

Answer

The middle term in the binomial expansion of \(\left(x-\frac{1}{x}\right)^{6}\) is \(-20\).

Step 1: Identify the number of terms in the expansion

In a binomial expansion with a power of n, there are n+1 terms. In this case, the power is 6, so there will be 7 terms in the expansion.

Step 2: Find the middle term number

Since there are 7 terms, the middle term will be the 4th term in the binomial expansion.

Step 3: Apply binomial coefficients formula

For a binomial expansion, a term in the expansion can be represented as: $$ \binom{n}{k}a^{n-k}b^{k} $$ Where n is the power, k is the term number (starting from 0) and a and b are the terms in the binomial expression. In this case, n = 6, k = 3 (since we want the 4th term), \(a = x\) and \(b = -\frac{1}{x}\).

Step 4: Compute the middle term

Using this information, we can calculate the value of the middle term: $$ \binom{6}{3}x^{6-3}\left(-\frac{1}{x}\right)^{3} = C(6, 3)x^{3}\left(-\frac{1}{x}\right)^{3} $$ Where C(6, 3) represents the binomial coefficient. Binomial coefficient can be calculated as: $$ C(n,k) = \frac{n!}{k!(n-k)!} $$ So, for C(6, 3): $$ C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!}=\frac{720}{(6)(6)}=20 $$ Now, substitute the value of C(6, 3) back into the formula and compute the middle term: $$ 20x^{3}\left(-\frac{1}{x}\right)^{3} = 20x^{3}\left(-\frac{1}{x^3}\right) $$ The final middle term of the expansion is: $$ -20 $$

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that describes the expansion of powers of a binomial, which is an expression composed of two terms added together, such as \(a + b\). According to this theorem, any positive integral power of a binomial can be expressed as a sum of terms called binomial coefficients.

When we have a binomial expression of the form \(a + b\) raised to the power of \(n\), it expands to \(a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n-1}ab^{n-1} + b^n\). This sequence of terms is derived from the patterns of Pascal's Triangle or can be computed using the formula \( \binom{n}{k} \) where \( k \) ranges from 0 to \( n \).

Binomial Coefficients

Binomial coefficients are the numerical factors that multiply the terms in the expansion of a binomial expression. These coefficients can be found in Pascal's Triangle or calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
where \( n \) is the power to which the binomial is raised, \( k \) refers to the specific term in the expansion (starting from 0), \( n! \) denotes \( n \) factorial, and \( (n-k)! \) is the factorial of the difference between \( n \) and \( k \).

For instance, the coefficient \( \binom{n}{0} \) is always 1 because any number raised to the power of 0 is 1, and the coefficient \( \binom{n}{n} \) is also 1 since we're multiplying the term by \( b^n \) only. Understanding these coefficients is vital for determining the terms of a binomial expansion, particularly if we seek to find a specific term, like the middle term in the exercise provided.

Middle Term of Binomial Expansion

Identifying the middle term of a binomial expansion requires understanding the symmetrical nature of binomial coefficients. When a binomial expression is raised to an even power \(n\), there are \(n+1\) terms, which means the middle term is the \(\frac{n}{2} + 1\)th term. If \(n\) is odd, there are two middle terms, namely the \(\frac{n+1}{2}\)th and the \(\frac{n+1}{2} + 1\)th terms.

In our exercise with \(\left(x-\frac{1}{x}\right)^{6}\), the power is 6 (even), thus providing us with 7 terms. The middle term is the 4th term (as in Step 2 of the solution). To find this term, we apply the binomial coefficients formula to obtain it, precisely like in Step 4 of the original solution. Knowing how to determine the middle term is crucial for solving problems related to binomial expansion and can also be helpful in probability and statistical applications where such distributions are used.