Middle Term in Binomial Theorem

The middle term in the binomial theorem plays a significant role in understanding the expansion of expressions of the form ${\left(a+b\right)}^{n}$. It is especially useful when n is an even number. In this context, the middle term refers to the term in the binomial expansion that appears at the center of the series. Let's explore the concept of the middle term in the binomial theorem.

Finding the Middle Term:

When the exponent 'n' in ${\left(a+b\right)}^{n}$ is even, the middle term can be found using the following formula:

Middle Term (M.T.) = C(n, n/2) * (${a}^{\frac{n}{2}}$) * (${b}^{\frac{n}{2}}$)

Where:

• M.T. represents the middle term.
• C(n, n/2) is the binomial coefficient, which calculates the number of ways to choose 'n/2' elements from a set of 'n' elements.
• 'a' and 'b' are the constants in the original expression (a + b).
• 'n' is the even exponent to which the binomial is raised.

To determine the middle term, we consider two cases:

1. When 'n' is an even integer: In this case, there are two middle terms because there is no single middle term. The middle terms are the ${\frac{n}{2}}^{th}$ and ${\left(\frac{n}{2}+1\right)}^{th}$terms in the expansion.

2. When 'n' is an odd integer: In this case, there is a single middle term, which is the${\frac{\left(n+1\right)}{2}}^{th}$term in the expansion.

Finding the Middle Term Formula:

To find the middle term(s) in the expansion of ${\left(a+b\right)}^{n}$, you can use the following formulas:

1. When 'n' is even (n = 2m for some integer 'm'):

Middle terms: T(m) and T(m+1)

• T(m) = C(n, m) * (${a}^{\left(n-m\right)}$) * (${b}^{m}$)
• T(m+1) = C(n, m+1) * (${a}^{\left(n-m-1\right)}$) * (${b}^{\left(m+1\right)}$)
2. When 'n' is odd (n = 2m + 1 for some integer 'm'):

Middle term: T(m+1)

• T(m+1) = C(n, m+1) * (${a}^{\left(n-m-1\right)}$) * (${b}^{\left(m+1\right)}$)

Key Points:

1. Even vs. Odd 'n': Whether 'n' is even or odd determines whether there is one or two middle terms in the expansion.

2. Binomial Coefficients: The binomial coefficients (C(n, k)) play a crucial role in calculating the middle terms, just like in finding the general terms.

3. Exponent Rules: The exponents of 'a' and 'b' in the middle terms depend on 'm' and 'n' as described in the formulas above.

Examples:

Let's consider a few examples to illustrate the concept of the middle term in the binomial theorem.

1. For ${\left(x+1\right)}^{4}$the middle term is found as follows:

• Middle Term (M.T.) = C(4, 4/2) * ${x}^{\frac{4}{2}}$ * ${1}^{\frac{4}{2}}$ = C(4, 2) * ${x}^{2}$ * ${1}^{2}$ = 6${x}^{2}$
2. For ${\left(a-2b\right)}^{6}$, the middle term can be determined as:

• Middle Term (M.T.) = C(6, 6/2) * ${a}^{\frac{6}{2}}$ * ${\left(-2b\right)}^{\frac{6}{2}}$ = C(6, 3) * ${a}^{3}$ * 4${b}^{3}$ = 20${a}^{3}$ * 4${b}^{3}$ = 80${a}^{3}$${b}^{3}$

3. For ${\left(x+y\right)}^{5}$, which has an odd 'n' (n = 5):

• The middle term is T(3).
• T(3) = C(5, 3) * ${x}^{2}$ * ${y}^{3}$ = 10${x}^{2}$${y}^{3}$

4. For ${\left(x+y\right)}^{4}$, which has an even 'n' (n = 4):

The middle terms are T(2) and T(3).

• T(2) = C(4, 2) * ${x}^{2}$ * ${y}^{2}$ = 6${x}^{2}$${y}^{2}$
• T(3) = C(4, 3) * x * ${y}^{3}$ = 4x${y}^{3}$