Middle Terms in Binomial Expansion:
∎ When n is even
Middle term of the expansion is , $\Large (\frac{n}{2} + 1)^{th} term $
∎ When n is odd
In this case $\large (\frac{n+1}{2})^{th} term $ term and$\large (\frac{n+3}{2})^{th} term $ are the middle terms.
e.g. Middle term in the expansion of (1 + x)4 and (1 + x)5
Expansion of (1 + x)4 has 5 terms, so third term is the middle term .
Expansion of (1 + x)5 has 6 terms, so 3rd and 4th both are middle terms .
Illustration : Find the middle term in the expression of
(1 − 2x + x2)n
Solution : (1− 2x + x2)n = [ ( 1 − x)2]n = ( 1− x)2n
Here 2n is even integer, therefore, (2n/2 + 1) th
i.e. (n + 1)th term will be the middle term.
Now (n + 1)th term in (1 − x)2n
= 2nCn (1)2n−n(−x)n = 2nCn(−x)n
$ \large = \frac{2n!}{n! n!}(-x)^n $
Greatest Binomial Coefficient :
Let the (r + 1)th term contains the greatest binomial co-efficient, then
$ \large \frac{T_{r+1}}{T_r} = \frac{C_r}{C_{r+1}}= \frac{n-r+1}{r}$
$ \large = \frac{n+1}{r}-1 > 1$
$ \large \frac{n+1}{r} > 2 $
$ \large \frac{n+1}{2} > r $ …(1)
But r must be an integer, and therefore when n is even,
the greatest binomial coefficient is given by the greatest value of r , consistent with (1)
i.e., r = n/2 and hence the greatest binomial coefficient is nCn/2
Similarly if n be odd, the greatest binomial coefficient is given when,
$ \large r = \frac{n-1}{2} \; or \; \frac{n+1}{2}$
And the coefficients will be $\large n_C{_{(n+1)/2}}$ and $\large n_C{_{(n-1)/2}}$ both being equal .
Note: The greatest binomial coefficient is the binomial coefficient of the middle term.
Illustration : Show that the greatest binomial co-efficient in the expansion of :
$\large (x+\frac{1}{x})^{2n}$ is $\large \frac{1.3.5….(2n-1).2^n}{n!}$
Solution: Since middle term has the greatest coefficient,
So, greatest coefficient = coefficient of middle term
= 2nCn
$\large = \frac{1.3.5….(2n-1).2^n}{n!}$
Greatest Term:
To determine the numerically greatest term (absolute value) in the expansion of (a + x)n , when n is a positive integer.
Consider ,
$\large |\frac{T_{r+1}}{T_r}|= \frac{n_{C_r} a^{n-r} x^r}{n_{C_{r-1}} a^{n-r+1} x^{r-1}}$
$\large = |\frac{n_{C_r}}{n_{C_{r-1}}}||\frac{x}{a}|$
$\large = |\frac{n-r+1}{r}||\frac{x}{a}|$
$\large = |\frac{n+1}{r}-1 ||\frac{x}{a}|$
Thus |Tr + 1| > |Tr| if
$\large (\frac{n+1}{r}-1 )|\frac{x}{a}| > 1$
Note:
∎ $\large (\frac{n+1}{r}-1 )|$ must be positive since n > r
ThusTr+1 will be the greatest term if, r has the greatest value consistent with the inequality
Illustration : Find the greatest term in the expansion of (2 + 3x)9 if x = 3/2
Solution:
$\large \frac{T_{r+1}}{T_r} = \frac{n-r+1}{r}(\frac{3x}{2})$
$\large = \frac{10-r}{r}(\frac{3x}{2})$ ;when x=3/2
$\large = \frac{10-r}{r}(\frac{3}{2})(\frac{3}{2})$
$\large = \frac{10-r}{r}(\frac{9}{4})$
$\large \frac{T_{r+1}}{T_r} = \frac{90-9r}{4r} $
Therefore Tr+1 ≥ Tr if,
90 – 9r ≥ 4r => 90 ≥ 13r
r ≤ 90/13 , r being an integer, hence r = 6
Tr+1 = T7 = T6+1 = 9C6 (2)3 (3x)6
Problems based on Greatest Term:
Illustration . Find the greatest term in the expansion of
$\large \sqrt{3}(1 + \frac{1}{\sqrt{3}})^{20}$
Solution: Let rth term be the greatest term.
Since , $\large \frac{T_r}{T_{r+1}} = \frac{r}{21-r}\times \sqrt{3}$
Now , $\large \frac{T_r}{T_{r+1}} \ge 1 $
$\large r \ge \frac{21}{\sqrt{3}+1}$ . . . . (1)
Now again ,
$\large \frac{T_{r-1}}{T_r} = \frac{r-1}{22-r}\times \sqrt{3} \le 1 $
$\large r \le \frac{22 + \sqrt{3}}{\sqrt{3}+1}$ . . (2)
From (1) and (2) follows that
$\large \frac{21}{\sqrt{3}+1} \le r \le \frac{22 + \sqrt{3}}{\sqrt{3}+1}$
=> r = 8 is the greatest term and its value
$\displaystyle = 20_{C_7}.\frac{1}{27}$
Properties of Binomial Expansion
∎ There are (n + 1) terms in the expansion of (a + b)n , the first and the last term being an and bn respectively.
∎ $\large \frac{T_{r+1}}{T_r}= \frac{n-r+1}{r} \frac{x}{a}$ for the binomial expansion (a + x)n
∎ Dm (ax + b)n = 0 if m > n where Dm is mth derivative w. r. t. x .
∎ Dm (ax + b)n = ann! if n = m
∎ Dm (ax + b)n $\large = \frac{a^m \; n!}{(n-m)!}(ax+b)^{n-m}$ ; (m < n)
Also Read :
| → Binomial Theorem : General Term in the Expansion → Properties of Binomial coefficients → Problems Related to Binomial coefficients → Binomial Theorem for any index |