Name: Finite Series in Binomial Coefficients
Uploaded: 2019-07-19
Duration: 6 min 12 s
Description: Learn the concept of summation of finite series in binomial coefficients and explore the use of derivative to find the sum of series as well by watching this...

Question

Show that if the greatest term in the expansion of

(1 + x)^{2 n}

also has the greatest coefficient, then

' x'

lies between,

\frac{n}{n + 1} & \frac{n + 1}{n}

Accepted Answer

Total number of terms are $2 n + 1$ in the binomial expansion of ${(1 + x)}^{2 n},$

Middle term is ${(\frac{2 n}{2} + 1)}^{t h} = {(n + 1)}^{t h}$

We know that, middle term has the greatest coefficient

$\Rightarrow T_{n} < T_{n + 1} > T_{n + 2}$

$\Rightarrow \frac{T_{n}}{T_{n + 1}} < 1 or \frac{T_{n + 1}}{T_{n + 2}} > 1$

$\Rightarrow \frac{T_{n}}{T_{n + 1}} = \frac{{}^{2 n}C_{n - 1} \cdot x^{n - 1}}{{}^{2 n}C_{n} \cdot x^{n}} < 1$

$\Rightarrow \frac{(2 n)! \cdot n! \cdot (2 n - n)!}{(2 n - n + 1)! \cdot (n - 1) \cdot (2 n)! \cdot x} < 1$

$\Rightarrow \frac{n}{(n + 1) \cdot x} < 1$

$\Rightarrow x > \frac{n}{n + 1} . . . . . (1)$

$\Rightarrow \frac{T_{n + 1}}{T_{n + 2}} = \frac{{}^{2 n}C_{n} \cdot x^{n}}{{}^{2 n}C_{n + 1} \cdot x^{n + 1}} > 1$

$\Rightarrow \frac{{}^{2 n}C_{n} \cdot x^{n}}{{}^{2 n}C_{n + 1} \cdot x^{n + 1}} = \frac{(2 n)! \cdot (2 n - n - 1)! \cdot (n + 1)!}{(2 n)! \cdot (2 n - n)! \cdot n! \cdot x} > 1$

$\Rightarrow \frac{n + 1}{n \cdot x} > 1$

$\Rightarrow x < \frac{n + 1}{n} . . . . . . . (2)$

By using equation $(1)$ and equation $(2)$ we get,

$\frac{n}{n + 1} < x < \frac{n + 1}{n} .$

Show that if the greatest term in the expansion of $(1 + x)^{2 n}$ also has the greatest coefficient, then $' x'$ lies between, $\frac{n}{n + 1} & \frac{n + 1}{n}$

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Show that if the greatest term in the expansion of (1+x)2n also has the greatest coefficient, then 'x' lies between, nn+1&n+1n

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Show that if the greatest term in the expansion of $(1 + x)^{2 n}$ also has the greatest coefficient, then $' x'$ lies between, $\frac{n}{n + 1} & \frac{n + 1}{n}$