Find the coefficient of x n in ( x 2 + x 3 + x...

Jaqueline Kirby

Jaqueline Kirby

Answered

2022-06-25

Find the coefficient of x n in ( x 2 + x 3 + x 4 + ) 5 .
I have got stuck on this question, though I realise that I have probably got really close to an answer. This is how I approached it:
f ( x ) = ( x 2 + x 3 + x 4 + ) 5 = x 10 ( 1 + x + x 2 + ) 5 = x 10 ( 1 x m 1 x ) 5 = x 10 ( 1 x m ) 5 ( 1 x ) 5 .
Then I have used the binomial theorem:
( 1 x m ) 5 = i = 0 5 ( 5 i ) ( 1 ) i ( x m ) i , ( 1 x ) 5 = j = 0 ( 4 + j j ) ( x ) j .
Therefore, f ( x ) = x 10 ( i = 0 5 ( 5 i ) ( 1 ) i ( x m ) i ) ( j = 0 ( 4 + j j ) x j ) ,
and as I work out coefficient of x n I arrive to:
[ x n ] = ( 5 0 ) ( n 6 n 10 )
I think my answer is incomplete and unfortunately, I don't have a solution to check it. I would really appreciate your help. Thank you

Answer & Explanation

jmibanezla

jmibanezla

Expert

2022-06-26Added 17 answers

Step 1
As you had before, we have
f ( x ) = ( x 2 + x 3 + ) 5
f ( x ) = x 10 ( 1 + x + ) 5
f ( x ) = x 10 ( 1 x ) 5
f ( x ) = x 10 ( 1 x ) 5
f ( x ) = x 10 k = 0 ( k + 4 4 ) x k
f ( x ) = k = 0 ( k + 4 4 ) x k + 10
Step 2
Hence, the coefficient of x n is ( n 6 4 ) .

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