The sum of the coefficients of all even degree terms is x in the expansion of $$x+x3$$−$$16+x$$−$$x3$$−$$16$$ (x$$$$1) is equa to

Question

The sum of the coefficients of all even degree terms is x in the expansion of $$x+x3$$−$$16+x$$−$$x3$$−$$16$$ (x$$>$$1)  is equa to

Infinity Learn · Accepted Answer

Given expression is $$x+x3$$−$$16+x$$−$$x3$$−$$16=2$$ $$6C0x6+6C2x4x3$$−$$12+6C4x2x3$$−$$14+6C6x3$$−$$16$$∵$$(a+b)n+(a$$−$$b)n=2$$ $$nC0an+nC2an$$−$$2b2+nC4an$$−$$4b4+$$… $$6C0x6+6C2x4x3$$−$$1+6C4x2x3$$−$$12+6C6x3$$−$$13$$ The sum of the terms with even power of $$x=2$$ $$6C0x6+6C2$$−$$x4+6C4x8+6C4x2+6C6$$−$$1$$−$$3x6$$ $$=2$$ $$6C0x6$$−$$6C2x4+6C4x8+6C4x2$$−$$1$$−$$3x6Now$$, the required sum of the coefficients of even powers of x in $$x+x3-16+x$$−$$x3-16=2$$ $$6C0$$−$$6C2+6C4+6C4$$−$$1$$−$$3=2$$[$$1$$−$$15+15+15$$−$$1$$−$$3$$]$$=2(15$$−$$3)=24$$

The sum of the coefficients of all even degree terms is x in the expansion of ${(x + \sqrt{x^{3} - 1})}^{6} + (x - \sqrt{{x^{3} - 1)}^{6}}$ $(x > 1)$ is equa to

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The sum of the coefficients of all even degree terms is x in the expansion of x+x3−16+x−x3−16 (x>1) is equa to

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The sum of the coefficients of all even degree terms is x in the expansion of ${(x + \sqrt{x^{3} - 1})}^{6} + (x - \sqrt{{x^{3} - 1)}^{6}}$ $(x > 1)$ is equa to