Middle Term and Properties of Binomial Coefficients

Finding the Middle Term

The total number of terms in the expansion of $(x+a)^n$ is $n+1$. The middle term depends on whether n is even or odd.

• If n is Even: The number of terms ($n+1$) is odd. There is one middle term, which is the $(\frac{n}{2} + 1)^{th}$ term.

• If n is Odd: The number of terms ($n+1$) is even. There are two middle terms: the $(\frac{n+1}{2})^{th}$ term and the $(\frac{n+1}{2} + 1)^{th}$ term.

Properties of Binomial Coefficients ($^nC_r$)

Binomial coefficients have several important properties that are frequently used in problem-solving.

1. Symmetry: The coefficient of the $k^{th}$ term from the beginning is equal to the coefficient of the $k^{th}$ term from the end.

$$^nC_r = ^nC_{n-r}$$

2. Pascal's Rule:

$$^nC_r + ^nC_{r-1} = ^{n+1}C_r$$

3. Sum of Coefficients: By setting $x=1$ in $(1+x)^n$, we find the sum of all binomial coefficients.

$$^nC_0 + ^nC_1 + ^nC_2 + ... + ^nC_n = 2^n$$

4. Sum of Even and Odd Coefficients:

$$^nC_0 + ^nC_2 + ^nC_4 + ... = ^nC_1 + ^nC_3 + ^nC_5 + ... = 2^{n-1}$$

5. Ratio of Consecutive Coefficients:

$$\frac{^nC_r}{^nC_{r-1}} = \frac{n-r+1}{r}$$

6. Greatest Binomial Coefficient: The value of $^nC_r$ is maximum at the middle term(s).
   • If n is even, the greatest coefficient is $^nC_{n/2}$.
   • If n is odd, the two greatest coefficients are $^nC_{(n-1)/2}$ and $^nC_{(n+1)/2}$ (and they are equal).

Example 1: Middle Term (n is Even)

Question: Find the middle term in the expansion of $(2x^2 - \frac{1}{x})^8$.

Solution: Here, n=8 (even). The number of terms is 9. The middle term is the $(\frac{8}{2}+1)^{th} = 5^{th}$ term. For the 5th term, $r=4$. Using the general term formula $T_{r+1} = {^nC_r}x^{n-r}a^r$:

$$T_5 = {^8C_4}(2x^2)^{8-4}(-\frac{1}{x})^4 = {^8C_4}(2x^2)^4(\frac{1}{x^4})$$

$$= 70 \cdot (16x^8) \cdot (\frac{1}{x^4}) = 1120x^4$$

Example 2: Middle Terms (n is Odd)

Question: Find the middle terms in the expansion of $(3x - \frac{x^3}{6})^7$.

Solution: Here, n=7 (odd). There are two middle terms: the $(\frac{7+1}{2})^{th}=4^{th}$ term and the $(\frac{7+1}{2}+1)^{th}=5^{th}$ term.

• For the 4th term ($r=3$):

$$T_4 = {^7C_3}(3x)^{7-3}(-\frac{x^3}{6})^3 = 35(3^4x^4)(-\frac{x^9}{216}) = -\frac{105}{8}x^{13}$$

• For the 5th term ($r=4$):

$$T_5 = {^7C_4}(3x)^{7-4}(-\frac{x^3}{6})^4 = 35(3^3x^3)(\frac{x^{12}}{1296}) = \frac{35}{48}x^{15}$$

Example 3: Sum of Coefficients

Question: If the sum of the coefficients in the expansion of $(a+b)^n$ is 4096, find the greatest binomial coefficient in the expansion.

Solution: The sum of the coefficients is $(1+1)^n = 2^n$. We are given $2^n = 4096$. Since $4096 = 2^{12}$, we have $n=12$. The greatest binomial coefficient occurs at the middle term. Since n=12 is even, the greatest coefficient is $^{12}C_{12/2} = ^{12}C_6$.

$$^{12}C_6 = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 924$$

Example 4: Using the Ratio Property

Question: The coefficients of three consecutive terms in the expansion of $(1+x)^n$ are 165, 330, and 462. Find n.

Solution: Let the terms be for $r, r+1, r+2$. The coefficients are $^nC_{r-1}=165, ^nC_r=330, ^nC_{r+1}=462$. Using the ratio property:

1. $\frac{^nC_r}{^nC_{r-1}} = \frac{n-r+1}{r} = \frac{330}{165} = 2 \implies n-r+1=2r \implies n-3r=-1$.
2. $\frac{^nC_{r+1}}{^nC_r} = \frac{n-r}{r+1} = \frac{462}{330} = \frac{7}{5} \implies 5n-5r=7r+7 \implies 5n-12r=7$. Solving the two linear equations, we find $n=11$.