Question

if the coefficients of 1/x and x^2 are equal in the expression of (ax - b/x^2 )^8 find the relationship between a and b

Answer 1

$\fbox \pink{ Expression Of Binomial Theorem }$

${ (x+a)^{n} = ^{n}cox^{n}a^{0} + ^{n}c1x^{n-1}a^{1}}$

${ ^{n}C2x^{n-2}a^{2}x - - - - - - ^{n}cnx^{0}a^{n}}$

$\sf\underline\bold \purple{ where \: in }$

${ for \: n^{+}ve \: integral}$

${ (1+ax+bx^{2})(1-2x) ^{18}}$

${=11+ax+bx^{2})(^{18}co^{-18}c1}$

${(2x) +^{18}c2(2x^{2})- - - - - - - -)}$

$\sf\underline\bold \green{ Coefficients \: of }$

${x^{3} = ^{-18}C3(2)^{3}+^{18}c2(2)^{2}a-^{18}C1(2)b=0}$

Coefficients of

${ x^{4}=^{-18}c4(2)^{4}+^{18}C3(2)^{3}a+^{18}C2(2)^{2}b = 0}$

${ —› -18×17×16×8/3×2+18×17}$

${ ×4a/2-18×2b=0}$

And

${ —› 18×17×16×15×16/4×3×2+18}$

${×17×16×8a6-18×17/2×4b=0}$

${—›12(-544+51a-3b)=0}$

${and \: 12×17(240-32a+3b)=0}$

${ a=16 }$

${ b=272/3 }$

$\large\sf\underline\bold \red{ Hope \: this \: will \: help \: you }$