A) \[{{5}^{n}}-{{4}^{n}}\]
B) \[{{5}^{n+1}}-{{4}^{n+1}}\]
C) \[{{5}^{n-1}}-{{4}^{n-1}}\]
D) None of these
Correct Answer: B
Solution :
We have, \[{{(1-9x+20{{x}^{2}})}^{-1}}={{[(1-5x)(1-4x)]}^{-1}}\] \[=\frac{1}{(1-5x)(1-4x)}=\frac{5}{1-5x}-\frac{4}{1-4x}\] \[=5{{(1-5x)}^{-1}}-4{{(1-4x)}^{-1}}\] \[=5[1+5x+{{(5x)}^{2}}+...+{{(5x)}^{n}}+...]\]\[-4[1+4x+{{(4x)}^{2}}.....+{{(4x)}^{n}}+...]\] Therefore the coefficient of \[{{x}^{n}}\] in \[{{(1-9x+20{{x}^{2}})}^{-1}}=5({{5}^{n}})-4({{4}^{n}})={{5}^{n+1}}-{{4}^{n+1}}\]
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