The coefficient of x^4 in the expansion of fr | vedclass

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(B) We use the expansion $(1-ax)^{-1} = 1 + ax + a^2x^2 + a^3x^3 + a^4x^4 + \dots$ The given expression is $(1-x)^{-1}(1-2x)^{-1}(1-3x)^{-1}$. Expandi

Vedclass · Accepted Answer

$301$

Vedclass · Answer

(B) We use the expansion $(1-ax)^{-1} = 1 + ax + a^2x^2 + a^3x^3 + a^4x^4 + \dots$ The given expression is $(1-x)^{-1}(1-2x)^{-1}(1-3x)^{-1}$. Expanding each term up to $x^4$: $(1+x+x^2+x^3+x^4)(1+2x+4x^2+8x^3+16x^4)(1+3x+9x^2+27x^3+81x^4)$. Multiplying these series,the coefficient of $x^4$ is obtained by summing all combinations of terms whose powers add up to $4$: $1(1)(81) + 1(2)(27) + 1(4)(9) + 1(8)(3) + 1(16)(1) + 1(3)(27) + 1(9)(8) + 1(27)(4) + 1(1)(27) + 1(3)(8) + 1(9)(4) + 1(27)(2) + 1(1)(16) + 1(3)(4) + 1(9)(2) + 1(27)(1) + 1(1)(9) + 1(3)(2) + 1(9)(1) + 1(1)(3) + 1(3)(1) + 1(1)(1) = 301$. Thus,the coefficient of $x^4$ is $301$.

The coefficient of $x^4$ in the expansion of $\frac{1}{(1-x)(1-2x)(1-3x)}$ is

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