Question

The coefficient of $x^{25}$ in $(1 + x + x^{2} + x^{3} + x^{4})^{- 1}$ is

• A. 25
• B. – 25
• C. 1
• D. – 1

Answer 1

Answer: (C)

1

Answer 2

Coefficient of $x^{25}$ in $(1 + x + x^{2} + x^{3} + x^{4})^{- 1}$

= Coefficient of $x^{25}$ in $\left\lbrack \frac{1(1 - x^{5})}{1 - x} \right\rbrack^{- 1}$

= Coefficient of $x^{25}$ in $(1 - x^{5})^{- 1}.(1 - x)$

= Coefficient of $x^{25}$ in [$(1 - x^{5})^{- 1} - x(1 - x^{5})^{- 1}$]

=$\lbrack 1 + (x^{5})^{1} + (x^{5})^{2} + ......\rbrack - x\lbrack 1 + (x^{5})^{1} + (x^{5})^{2}\rbrack + ......\rbrack$ = Coefficient of $x^{25}$ in $\lbrack 1 + x^{5} + x^{10} + x^{15} + .....\rbrack$

– Coefficient of $x^{24}$ in $\lbrack 1 + x^{5} + x^{10} + x^{15} + .....\rbrack$ = $1 - 0 = 1$.