First slide

Binomial theorem for positive integral Index

Question

If $O$ be the sum of odd terms and $E$ that of even terms in the expansion of $(x + a)^{n},$ then $O^{2} - E^{2} =$

Moderate

A

${(x^{2} + a^{2})}^{n}$
B

${(x^{2} - a^{2})}^{n}$
C

$(x - a)^{2 n}$
D

none of these

Solution

we have ,
$(x + a)^{n} =^{n} C_{0} x^{n} a^{0} +^{n} C_{1} x^{n - 1} a^{1} +^{n} C_{2} x^{n - 2} a^{2} + \dots +^{n} C_{n - 1} x a^{n - 1} +^{n} C_{n} a^{n}$
$\Rightarrow (x + a)^{n} = (^{n} C_{0} x^{n} a^{0} +^{n} C_{2} x^{n - 2} a^{2} + \dots) + (^{n} C_{1} x^{n - 1} a^{1} +^{n} C_{3} x^{n - 3} a^{3} + \dots)$
$\Rightarrow (x + a)^{n} = O + E$ (i)
and ,
$\begin{array}{l} (x - a)^{n} =^{n} C_{0} x^{n} -^{n} C_{1} x^{n - 1} a^{1} +^{n} C_{2} x^{n - 2} a^{2} -^{n} C_{3} x^{n - 3} a^{3} + \dots \\ +^{n} C_{n - 1} x (- 1)^{n - 1} a^{n - 1} +^{n} C_{n} (- 1)^{n} a^{n} \\ \Rightarrow (x - a)^{n} = (^{n} C_{0} x^{n} +^{n} C_{2} x^{n - 2} a^{2} + \dots) - (^{n} C_{1} x^{n - 1} a^{1} +^{n} C_{3} x^{n - 3} a^{3} + \dots) \end{array}$
$\Rightarrow (x - a)^{n} = O - E$ (ii)
From (i) and (ii), we get
$\begin{array}{l} (x + a_{i}^{n} (x - a)^{n} = (O + E) (O - E) \\ \Rightarrow {(x^{2} - a^{2})}^{n} = O^{2} - E^{2} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App