The coefficient of xn in the expansion of (1-9x+20x2)-1

Question

The coefficient of xn in the expansion of (1-9x+20x2)-1  (A)  5n-4n  (B)  5n+1-4n+1  (C)  5n-1-4n-1  (D) None of these

Tardigrade Admin · Accepted Answer

We have, (1-9 x+20 x2)-1=[(1-5 x)(1-4 x)]-1 =(1/(1-5 x)(1-4 x))=(5/1-5 x)-(4/1-4 x) =5(1-5 x)-1-4(1-4 x)-1 =5[1+5 x+(5 x)2+ ldots+(5 x)n+ ldots] -4[1+4 x+(4 x)2+ ldots+(4 x)n+ ldots] ⇒ The coefficient of xn is 5n+1-4n+1

Q. The coefficient of $x^{n}$ in the expansion of $(1 - 9 x + 20 x^{2})^{- 1}$

$5^{n} - 4^{n}$

$5^{n + 1} - 4^{n + 1}$

$5^{n - 1} - 4^{n - 1}$

None of these

Q. The coefficient of xn in the expansion of (1−9x+20x2)−1

5n−4n

5n+1−4n+1

5n−1−4n−1

None of these

We have, (1−9x+20x2)−1=[(1−5x)(1−4x)]−1 =(1−5x)(1−4x)1​=1−5x5​−1−4x4​ =5(1−5x)−1−4(1−4x)−1 =5[1+5x+(5x)2+…+(5x)n+…] −4[1+4x+(4x)2+…+(4x)n+…]⇒ The coefficient of xn is 5n+1−4n+1

Q. The coefficient of $x^{n}$ in the expansion of $(1 - 9 x + 20 x^{2})^{- 1}$

$5^{n} - 4^{n}$

$5^{n + 1} - 4^{n + 1}$

$5^{n - 1} - 4^{n - 1}$