Question

Find the coefficient of ${x}^{n}$ in the expansion of

$\frac{2 + x + {x}^{2} }{ {(1 + x)}^{3} }$


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Answer 1

not sure tried to do the problem

Answer 2

Solution:-

Refer to the Attachment!!

Method of Solution:-

Step 1:

Just sent the ( 1+x)³ from Denominator to Numerator.

( 2 + x + x²)/(1+x)³

=> ( 2+x+x²)(1+x)^(-3).

Then,

Taken 1/2 common.

1/2[ 2 (x+1)(x+2)+x(x+1)+(x-1)x]

=> 2(-1)ⁿ/2 [ x² + 2x + x +2+2x² + 2x + 2x² -2x²]

After Cancelling we get,

=> (-1)ⁿ[ 5x² + 3x + 2]

Hence,

(-1)ⁿ[ 5x² + 3x + 2] is the Answer.