Question

Prove that the coefficient of x^n in the expansion (1+x+x2)^-1​ is 1,0,−1 according as n is of the form 3m,3m−1 or 3m+1​

Answer 1

Dear Student,

⇒Answer :

Let,

∴ y  =  $\frac{1}{ 1+ x + x^{2} }$

Multiply and divide by  1-x

y  =  $\frac{1 - x }{1 - x^{3} } = ( 1 - x ) [ 1 - x^{3} ] ^{-1}$

Expanding the later bracket

= ( 1 - x ) [ 1 + x³  +  2 × $\frac{ x^{6} }{2!}$  +  2 × 3  $\frac{ x^{ 9 } }{ 3 ! }$ ··· (−1)(−2) × (−3) × (−m)  $\frac{ -x^{3m} }{m!}$ ..  ]

= (1−x)  [1+ x³ + x ⁶ + x ⁹ .. $x^{ 3m-3}$  + $x^{ 3m }$... ]

coefficient of $x^{ 3 m }$

⇒ 1

coefficient of $x^{3m+1 }$

⇒ - 1

coefficient of $x^{ 3m -1 }$

⇒0  because no such term exits