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Question


Coefficient of xn in (1+x)(1+2x)(1+3x)(1−x)(1−2x)(1−3x) is

A
1230.2n20.3n
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B
1230.2n+20.3n
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C
12+30.2n+20.3n
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D
12+30.2n20.3n
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Solution

The correct option is B 1230.2n+20.3n
First we write the expression as sum of three terms using partial fraction.
To use partial fraction, first wirte the given expression as (1+x)(1+2x)(1+3x)(1x)(12x)(13x)=Ax+B1x+C12x+D13x
Now, simplpifying and comparing the cooefficients of x3 on both sides, we get
6=6AA=1
Similarly comparing the coefficients of x2 on both sides, we get
11=2A3A+6B+3C+2D
6B+3C+2D=16(1) (using A=1)
Comparing the coefficients of x on both sides, we get
6=A2B3B4C3D
5B4C3D=5(2)
Finally, comparing the constant terms, we get
B+C+D=1(3)
Solving (1),(2),(3) simultaneously, we get B=11,C=30,D=20
Hence, the given expression can be written as
x+111x+3012x+2013x
=(x+11)(1x)1+(30)(12x)1+20(13x)1
=(x+11)(1+x+x2+x3+...+xn1+xn+...)+(30)(1+2x+22x2+...2nxn+...)+20(1+3x+32x2+...+3nxn+...)
Hence, the coefficient of xn is equal to 1230.2n+20.3n

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