If coefficient of $$x3$$ and $$x4$$ in the expansion of $$1+ax+bx2(1$$−$$2x)18$$ in powers of x are both zeros, then (a$$,$$b) is equal to

Question

If coefficient of $$x3$$ and $$x4$$ in the expansion of $$1+ax+bx2(1$$−$$2x)18$$ in powers of x are both zeros, then (a$$,$$b) is equal to

Infinity Learn · Accepted Answer

$$S=1+ax+bx2(1$$−$$2x)18=1+ax+bx21+18C1($$−$$2x)+$$ $$18C2($$−$$2x)2+18C3($$−$$2x)3+18C4($$−$$2x)4+$$…Coefficient of $$x3$$ in the expansion of S is  $$18C3($$−$$2)3+a$$ $$18C2($$−$$2)2+18C1($$−$$2)b=0Divide$$ by  $$18C1($$−$$2)$$ to $$obtain5443$$−$$17a+b=0$$               (1)Similarly$$, coefficient of $$x4$$ is $$18C4($$−$$2)4+a$$ $$18C3($$−$$2)3+18C2($$−$$2)2b=0Divide$$ by  $$18C2($$−$$2)2$$ to $$obtain80$$−$$323a+b=0$$                 $$(2)Subtract$$ $$(2) from (1) to $$obtain3043$$−$$193a=0$$⇒$$a=16From$$ $$b=17$$×$$16$$−$$5443=2723$$

If coefficient of $x^{3}$ and $x^{4}$ in the expansion of $(1 + a x + b x^{2}) (1 - 2 x)^{18}$ in powers of x are both zeros, then $(a, b)$ is equal to

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If coefficient of x3 and x4 in the expansion of 1+ax+bx2(1−2x)18 in powers of x are both zeros, then (a,b) is equal to

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If coefficient of $x^{3}$ and $x^{4}$ in the expansion of $(1 + a x + b x^{2}) (1 - 2 x)^{18}$ in powers of x are both zeros, then $(a, b)$ is equal to