Problem 11 - Discrete Mathematics with Applications

Question

Find the middle term in the binomial expansion of each. $$\left(2 x+\frac{2}{x}\right)^{8}$$

Step-by-Step Solution

Answer

The middle term in the binomial expansion of \(\left(2x + \frac{2}{x}\right)^8\) is \(1792x^2 + 1120\).

Step 1: Understand the binomial expansion formula

The binomial expansion formula is given by: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] where \(a\) and \(b\) are terms inside the binomial expression, \(n\) is the power, \(\binom{n}{k}\) represents the binomial coefficient, and the summation runs from \(k = 0\) to \(k = n\).

Step 2: Determine the binomial coefficients for the middle terms

In our given expression, \[n = 8\], and when the power is even, there will be two middle terms (with indices \(\frac{n}{2} - 1\) and \(\frac{n}{2}\)). Now, the binomial coefficients can be calculated using: \[\binom{n}{k} = \frac{n!}{(n-k)!k!}\] The two middle terms for our expression will have the indices: \(k_1 = \frac{8}{2} - 1 = 3\), and \(k_2 = \frac{8}{2} = 4\) We will first calculate \(\binom{8}{3}\) and \(\binom{8}{4}\). \[\binom{8}{3} = \frac{8!}{(8-3)!3!} = \frac{8!}{5!3!} = 56\] \[\binom{8}{4} = \frac{8!}{(8-4)!4!} = \frac{8!}{4!4!} = 70\]

Step 3: Calculate the middle terms using the binomial expansion formula

Now, we will use the calculated binomial coefficients and the binomial expansion formula to find the middle terms. Middle term 1 with index \(k_1 = 3\): \[\binom{8}{3} (2x)^{8-3} \left(\frac{2}{x}\right)^3 = 56 (2^5 x^5) (2^3 x^{-3})\] Middle term 2 with index \(k_2 = 4\): \[\binom{8}{4} (2x)^{8-4} \left(\frac{2}{x}\right)^4 = 70 (2^4 x^4) (2^4 x^{-4})\]

Step 4: Combine the middle terms and simplify

Now, let's combine the two middle terms and simplify the expression. Middle term = Middle term 1 + Middle term 2 \[ = 56 (2^5 x^5) (2^3 x^{-3}) + 70 (2^4 x^4) (2^4 x^{-4})\] \[ = 56 (32x^2) + 70 (16x^0)\] \[ = 56(32x^2) + 70(16)\] \[ = 1792x^2 + 1120\] Now, we have found the middle term in the binomial expansion: \[1792x^2 + 1120\]

Key Concepts

Binomial Coefficients

In mathematics, binomial coefficients are a fundamental element of the binomial expansion. They describe how many ways there are to choose a subset of elements from a larger set.
They are commonly written as \( \binom{n}{k} \), pronounced "n choose k," and are defined by the formula:
\[\binom{n}{k} = \frac{n!}{(n-k)!k!}\]where \(n!\) represents the factorial of \(n\), which is the product of all positive integers up to \(n\).
This concept allows us to determine the coefficients of each term in a binomial expansion.

• For example, in the expansion of \((a + b)^8\), the coefficient of the term \(a^k b^{8-k}\) is given by \(\binom{8}{k}\).
• In our exercise, the coefficients of the middle terms, which are \(\binom{8}{3}\) and \(\binom{8}{4}\), are calculated as 56 and 70, respectively.

These coefficients are critical for calculating the specific terms in any binomial expansion.

Middle Term in Binomial Expansion

To find the middle term in a binomial expansion, particularly when the exponent is even, we look for the terms around the center of the expansion. The middle term(s) depend on the value of \(n\), the exponent of the binomial.

• When \(n\) is an even number, there are two middle terms, which are located at positions \(\frac{n}{2} - 1\) and \(\frac{n}{2}\).
• For our given binomial \((2x + \frac{2}{x})^8\), these indices are 3 and 4, because \(\frac{8}{2} - 1 = 3\) and \(\frac{8}{2} = 4\).
• These indices help us calculate the middle terms using the binomial expansion formula.

The middle terms in the expansion give us insight into the configuration of the expression at its center, which can sometimes be symmetrical or have unique properties like specific powers of variables, as seen in \(1792x^2 + 1120\). This gives key insights into the nature of the binomial expression.

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions that are raised to a power into simpler terms. The theorem provides a way to write the expansion of \((a + b)^n\) using binomial coefficients.
The general formula for binomial expansion is:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]This sum goes from \(k = 0\) to \(k = n\), covering all terms in the expansion. The theorem is incredibly useful for simplifying complex binomial expressions.

• It provides a systematic way to expand expressions without manually multiplying the terms repeatedly.
• Each term in the expansion consists of the product of the binomial coefficient, and the terms \(a\) and \(b\) raised to appropriate powers that sum to \(n\).
• In our exercise, the binomial theorem is employed to expand the expression \((2x + \frac{2}{x})^8\) into individual terms, focusing particularly on finding the middle terms.

Employing the binomial theorem facilitates efficient and accurate expansions, proving invaluable in both pure and applied mathematics.