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Hi.

I am confused with this question. I tried two different ways to solve it, but I got different answers for each way. The question is

"Determine the coefficient of x^100 in the pwer series form of (1+x+x^2)/((1-x^3)^2)"

First, I tried this way,

since we know 1/(1-x^3)=(1+x^3+x^6+x^9+.....), so the form becomes

(1+x+x^2)*(1+x^3+x^6+x^9+...)^2

let a0,a1 be the power of x from each series,then a0+a1=100.

from the first part we know that a0=0 or 1 or 2.

When a0=0, a1^2=a1+a1=100

when a0=1, a1^2=a1+a1=99

when a0=2, a1^2=a1+a1=98

and we know that a1 is multiple of 3, let a1=3n, then 6n=100.

Since we do not have n value, the coefficient of x^100 is 0.

Second I tried this way,

since we know 1+x+x^2=(1-x^3)/(1-x), if we apply this to our form, (1-x^3) from top and bottom is gone, therefore we have (1/(1-x))*(1/(1-x^3)).

since we know, 1/(1-x)=1+x+x^2+x^3+....

and 1/(1-x^3)=1+x^3+x^6+x^9+...

therefore we now have (1+x+x^2+x^3+...)*(1+x^3+x^6+x^9+...)

let a0 and a1 be the power of x from each series, then a0+a1=100.

Since a1 is the multiple of 3, it can be written as 3a1, then a0+3a1=100.

We have (a0,a1) as (1,33),(10,30),(19,27),(28,24)...(100,0) there are 12 sets

and each has coefficient of 1,

therefore the coefficient is 12.

I got two different answers. I don't know what is the problem...

Please tell me whether I am right or wrong...

Hi.

I am confused with this question. I tried two different ways to solve it, but I got different answers for each way. The question is

"Determine the coefficient of x^100 in the pwer series form of (1+x+x^2)/((1-x^3)^2)"

First, I tried this way,

since we know 1/(1-x^3)=(1+x^3+x^6+x^9+.....), so the form becomes

(1+x+x^2)*(1+x^3+x^6+x^9+...)^2

let a0,a1 be the power of x from each series,then a0+a1=100.

from the first part we know that a0=0 or 1 or 2.

When a0=0, a1^2=a1+a1=100

when a0=1, a1^2=a1+a1=99

when a0=2, a1^2=a1+a1=98

and we know that a1 is multiple of 3, let a1=3n, then 6n=100.

Since we do not have n value, the coefficient of x^100 is 0.

Second I tried this way,

since we know 1+x+x^2=(1-x^3)/(1-x), if we apply this to our form, (1-x^3) from top and bottom is gone, therefore we have (1/(1-x))*(1/(1-x^3)).

since we know, 1/(1-x)=1+x+x^2+x^3+....

and 1/(1-x^3)=1+x^3+x^6+x^9+...

therefore we now have (1+x+x^2+x^3+...)*(1+x^3+x^6+x^9+...)

let a0 and a1 be the power of x from each series, then a0+a1=100.

Since a1 is the multiple of 3, it can be written as 3a1, then a0+3a1=100.

We have (a0,a1) as (1,33),(10,30),(19,27),(28,24)...(100,0) there are 12 sets

and each has coefficient of 1,

therefore the coefficient is 12.

I got two different answers. I don't know what is the problem...

Please tell me whether I am right or wrong...

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