If O be the sum of odd terms and E that of even terms in the expansion of $$(x+a)n$$,then $$O2$$−$$E2=$$

Question

If O  be the sum of odd terms and  E that of even terms in the expansion of  $$(x+a)n$$,then $$O2$$−$$E2=$$

Infinity Learn · Accepted Answer

we have , (x+a)n=nC0xna0+nC1xn$$−$$1a1+nC2xn$$−$$2a2+$$…$$+nCn$$−$$1xan$$−$$1+nCnan$$⇒$$(x+a)n=$$ $$nC0xna0+nC2xn$$−$$2a2+$$…$$+$$ $$nC1xn$$−$$1a1+nC3xn$$−$$3a3+$$…⇒$$(x+a)n=O+E$$                           $$(i)and$$ , $$(x$$−$$a)n=nC0xn$$−$$nC1xn$$−$$1a1+nC2xn$$−$$2a2$$−$$nC3xn$$−$$3a3+$$…$$+nCn$$−$$1x($$−$$1)n$$−$$1an$$−$$1+nCn($$−$$1)nan$$⇒$$(x$$−$$a)n=$$ $$nC0xn+nC2xn$$−$$2a2+$$…− $$nC1xn$$−$$1a1+nC3xn$$−$$3a3+$$…⇒$$(x$$−$$a)n=O$$−E                          $$(ii)From$$ $$(i) and (ii), we get $$x+ain(x$$−$$a)n=(O+E)(O$$−$$E)$$⇒$$x2$$−$$a2n=O2$$−$$E2$$

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

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If O be the sum of odd terms and E that of even terms in the expansion of (x+a)n,then O2−E2=

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