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Prove that the coefficient of x^(n) in t...

Prove that the coefficient of `x^(n)` in the expansion of `(1)/((1-x)(1-2x)(1-3x))` is `1/2(3^(n+2)-2^(n+3)+1)`

Text Solution

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We have `(1)/((1-x)(1-2x)(1-3x))`
` = (1)/(2(1-x))- (4)/(1-2x)+(9)/(2(1-3x))` [By resolving into partial fractions]
`= 1/2(1-x)^(-1)-4(1-2x)^(-1)+9/2(1-3x)^(-1)`
`= 1/2(1+x+x^(2)+"….."+x^(n)+"…..")-4(1+2x+(2x)^(2)+"…."+(2x)^(n)+"….")+9/2(1+(3x)+(3x)^(2)+"...."+(3x)^(n)+"......")`
`:.` Coefficient of `x^(n) = 1/2[1-8.2^(n)+9.3^(n)]`
`=1/2[1-2^(n+3)+3^(n+2)]`
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