1. What is a Middle Term?
In the expansion of \((a + b)^n\), all terms appear in sequence from \(T_1\) to \(T_{n+1}\). The middle term is the one that appears exactly in the center of the list of terms.
The middle term is useful because it often has a special structure and is commonly needed in problems involving symmetry or central coefficients.
2. Case: \(n\) Even — One Middle Term
When \(n\) is even, there is one middle term in the expansion of \((a + b)^n\).
The total number of terms is \(n + 1\). If \(n\) is even, \(n + 1\) is odd, so there is a single center term.
2.1. Finding the Middle Term
If \(n\) is even, the middle term is:
\[ T_{\dfrac{n}{2} + 1} \]
Use the formula:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
2.2. Example (Even \(n\))
Find the middle term of \((x + 3)^8\).
Here, \(n = 8\) (even).
Middle term = \(T_{\dfrac{8}{2} + 1} = T_5\).
Compute:
\[ T_5 = \binom{8}{4} x^{8-4} 3^4 \]
- \(\binom{8}{4} = 70\)
- \(x^4\)
- \(3^4 = 81\)
So, middle term:
\[ T_5 = 70 \cdot 81 \cdot x^4 = 5670x^4 \]
3. Case: \(n\) Odd — Two Middle Terms
When \(n\) is odd, the number of terms is even (\(n + 1\)), so there are two middle terms.
These appear side by side in the center of the expansion.
3.1. Finding the Two Middle Terms
If \(n\) is odd, the middle terms are:
\[ T_{\dfrac{n+1}{2}} \quad \text{and} \quad T_{\dfrac{n+3}{2}} \]
3.2. Example (Odd \(n\))
Find the middle terms of \((2x - 1)^7\).
Here, \(n = 7\) (odd).
The middle terms are \(T_4\) and \(T_5\).
First Middle Term: \(T_4\)
\[ T_4 = \binom{7}{3} (2x)^{4} (-1)^3 \]
- \(\binom{7}{3} = 35\)
- \((2x)^4 = 16x^4\)
- \((-1)^3 = -1\)
\(T_4 = 35 \cdot 16x^4 \cdot (-1) = -560x^4\)
Second Middle Term: \(T_5\)
\[ T_5 = \binom{7}{4} (2x)^{3} (-1)^4 \]
- \(\binom{7}{4} = 35\)
- \((2x)^3 = 8x^3\)
- \((-1)^4 = 1\)
\(T_5 = 35 \cdot 8x^3 = 280x^3\)
4. Summary of Middle Term Locations
- If \(n\) is even → one middle term: \(T_{\dfrac{n}{2} + 1}\)
- If \(n\) is odd → two middle terms: \(T_{\dfrac{n+1}{2}}\) and \(T_{\dfrac{n+3}{2}}\)
This pattern works for any expansion of the form \((a + b)^n\).