First slide

Binomial theorem for positive integral Index

Question

If $P (x) = \frac{1}{\sqrt{3 x + 1}} [{(\frac{1 + \sqrt{3 x + 1}}{5})}^{n} - {(\frac{1 - \sqrt{3 x + 1}}{5})}^{n}]$ is a $5 t h$ degree polynomial, then value of $n$ is

Moderate

A

$9$
B

$11$
C

$23$
D

$21$

Solution

$P (x) = \frac{1}{5^{n} a} [(1 + a)^{n} - (1 - a)^{n}]$
where $a = \sqrt{3 x + 1}$
Now, $P (x) = \frac{2}{5^{n} a} [^{n} C_{1} a +^{n} C_{3} a^{3} + \dots]$
Note that n must be odd, so that $\sqrt{3 x + 1}$ does not appear in $P (x)$ .
$\begin{array}{l} ∴ P (x) = \frac{2}{5^{n}} [^{n} C_{1} +^{n} C_{3} a^{2} + \dots +^{n} C_{n} a^{n - 1}] \\ = \frac{2}{5^{n}} [^{n} C_{1} +^{n} C_{3} (3 x + 1) +^{n} C_{5} (3 x + 1)^{2} + \dots +^{n} C_{n} (3 x + 1)^{(n - 1) / 2}] \end{array}$
For this to be a polynomial of degree $5, \frac{n - 1}{2} = 5 \Rightarrow n = 11 .$

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