kenneth rosen

Question

$1)$FIND THE COEFFICIENT OF $x^{10}$ from $(x^3+x^4+x^5+x^6+x^7....)^3$

$2)$FIND THE COEFFICIENT OF $x^9$ from $(1+x+x^2)^3$

Comments

0 for 2nd one?

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I think in 1st question $x^{5}$ is missing.

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by mistake (x^3+x^4+x^5+x^6+x^7+........)^3

Answer 1

1) FIND THE COEFFICIENT OF X^10        (x^3+x^4+x^5+x^6+x^7....)^3

$\\ [(x^{3})^3](1+x+x^{2}+x^{3}+------)^{3}\\ =[x^{9}]( \sum_{k=0}^{\infty}x^{k})^{3} \\ = [x^{9}](\frac{1}{1-x})^{3} \\ =[x^{9}](\sum_{k=0}^{\infty}\binom{3+k-1}{k}x^{k}) \\ \\ Coefficient \,\, of \,\, x^{10}, \\\ = [x^{9}](\binom{3+k-1}{k}x^{k}) \\\ \\ put \, k=1, \\\ = \binom{3+1-1}{1}x^{10} \\ = \binom{3+1-1}{1} = 3$

 

 

2) FIND THE COEFFICIENT OF X^9   (1+x+x^2)^3

as you can see maximum power in this expansion is $x^{6}$ so coefficient  of $x^{9}$ will be 0.