If in the expansion of $$x3$$−$$1x2nthe$$ sum of the coefficients of $$x5$$ and $$x10$$ is $$0$$ then the coefficient of $$x20is$$

Question

If in the expansion of  $$x3$$−$$1x2nthe$$ sum of the coefficients of $$x5$$ and  $$x10$$ is  $$0$$ then the coefficient of $$x20is$$

Infinity Learn · Accepted Answer

(r$$ $$+$$ $$1)th$$ term in the expansion $$ofx3$$−$$1x2n$$ is $$Tr+1=nCrx3n$$−r−$$1x2r=nCrx3n$$−$$5r($$−$$1)rFor$$ coefficient of $$r10$$, set  $$3n$$−$$5r=10$$⇒ $$r=35n$$−$$1=r1$$  $$(say) Note that, $$r1=r2+1We$$ are given $$nCr1($$−$$1)r1+nCr2($$−$$1)r2=0$$ $$nCr2=nCr2+1$$∵$$r1=r2+1$$⇒$$r2+r2+1=n$$⇒$$r2=12(n$$−$$1)$$⇒ $$35n$$−$$2=12n$$−$$12$$⇒$$110n=32$$⇒$$n=15For$$ coefficient of $$x20$$ set  $$3n$$−$$5r=205r=45$$−$$20=25$$ or  $$r=5$$ Thus, coefficient of $$x20$$ is  −$$15C5$$

If in the expansion of ${(x^{3} - \frac{1}{x^{2}})}^{n}$ the
sum of the coefficients of $x^{5}$ and $x^{10}$ is $0$
then the coefficient of $x^{20}$ is

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If in the expansion of x3−1x2nthe sum of the coefficients of x5 and x10 is 0 then the coefficient of x20is

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Ready to Test Your Skills?

free study material

If in the expansion of ${(x^{3} - \frac{1}{x^{2}})}^{n}$ the
sum of the coefficients of $x^{5}$ and $x^{10}$ is $0$
then the coefficient of $x^{20}$ is