Prove that the coefficient of `x^n` in the expansion of `1/((1-x)(1-2x)(1-3x))i s1/2(3^(n+2)-2^(n+3)+1)dot`
Prove that the coefficient of `x^n` in the expansion of `1/((1-x)(1-2x)(1-3x))i s1/2(3^(n+2)-2^(n+3)+1)dot`
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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 [1-8xx2^(n)+9xx3^(n)]`
` = 1/2[1-2^(n+3)+3^(n+2)]`
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