Find the coefficient of xn in the expansion of 2-x-x21-x3

Given eqn can be written as y=(2+x+x^2)[1+x]^ 3 =(2+x+x^2)[1 3x+( 3)( 4).x^22!+. . . . . .. . . .( 3)( 4)...(n+1).x^n 1n 1! +( 3)( 4)...(n+2).x^nn! +. ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Middle Terms

Find the coef...

Question

Find the coefficient of $x^{n}$ in the expansion of $\frac{2 + x + x^{2}}{(1 + x)^{3}}$

Open in App

Solution

given eqn can be written as

$y = (2 + x + x^{2}) [1 + x]^{- 3}$

$= (2 + x + x^{2}) [1 - 3 x + \frac{(- 3) (- 4) . x^{2}}{2!} + . . . . . . . . . . \frac{(- 3) (- 4) . . . (n + 1) . x^{n - 1}}{n - 1!} + \frac{(- 3) (- 4) . . . (n + 2) . x^{n}}{n!} + . . . . . .]$

coefficient of $x^{n} = 2. (- 1)^{n - 2} . (n + 1) (n + 2) + (- 1)^{n - 2} . (n) (n + 1) + (- 1)^{n - 2} . (n - 1) (n)$

$= (- 1)^{n - 2} [4 n^{2} + 6 n + 4]$

Suggest Corrections

Similar questions

Q. Find the coefficient of

x

in the expansion of

(1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . . + \frac{x^{n}}{n!})

Q. If

| x | < 1

then the coefficient of

x^{n}

in expansion of

{(1 + x + x^{2} + x^{3} . . . .)}^{2}

Q. If

| x | < 1

, then the coefficient of

x^{n}

in the expansion

{(1 + x + x^{2} + x^{3} + . . .)}^{2}

Q. Find the coefficient of

x^{n}

in the expansion of

(\frac{1 + a x + x^{2}}{e^{x}})

Q. If

| x | < 1

,then the coefficient of

x^{n}

in the expansion of

(1 + x + x^{2} + x^{3} + . . . .)^{2}

Join BYJU'S Learning Program

Find the coefficient of xn in the expansion of 2+x+x2(1+x)3

given eqn can be written as y=(2+x+x2)[1+x]−3=(2+x+x2)[1−3x+(−3)(−4).x22!+..........(−3)(−4)...(n+1).xn−1n−1!+(−3)(−4)...(n+2).xnn!+......]coefficient of xn=2.(−1)n−2.(n+1)(n+2)+(−1)n−2.(n)(n+1)+(−1)n−2.(n−1)(n)=(−1)n−2[4n2+6n+4]

Find the coefficient of $x^{n}$ in the expansion of $\frac{2 + x + x^{2}}{(1 + x)^{3}}$