If the expansion in powers of x of the function  \dpi{100} \frac{1}{(1-ax)(1-bx)}is\, a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+.......,\, then\; a_{n}\; is

  • Option 1)

    \frac{b^{n}-a^{n}}{b-a}\;

  • Option 2)

    \; \; \frac{a^{n}-b^{n}}{b-a}\; \;

  • Option 3)

    \; \frac{a^{n+1}-b^{n+1}}{b-a}\; \;

  • Option 4)

    \; \; \frac{b^{n+1}-a^{n+1}}{b-a}\; \; \;

 

Answers (1)

As we learnt in

Properties of Binomial Theorem -

\left ( 1-x \right )^{-n}= 1+nx+\frac{n\left ( n+1 \right )}{2!}x^{2}+\frac{n\left ( n+1 \right )\left ( n+2 \right )}{3!}x^{3}+---
         

-

 

 Expansion of (1 - ax)-1 (1 - bx)-1 

\Rightarrow\ \; \left(1+ax+a^{2}x^{2}+a^{3}b^{3}+..........a^{n}b^{n} \right ) \left(1+bx+b^{2}x^{2}+.........+b^{n}x^{n} \right )

Coefficient of xn \Rightarrow\ \; a^{n}+a^{n-1}b+a^{n-2}b^{2}+........+b^{n}

\Rightarrow\ \; \frac{b^{n+1}-a^{n+1}}{b-a}

Correct option is 4.

 


Option 1)

\frac{b^{n}-a^{n}}{b-a}\;

This is an incorrect option.

Option 2)

\; \; \frac{a^{n}-b^{n}}{b-a}\; \;

This is an incorrect option.

Option 3)

\; \frac{a^{n+1}-b^{n+1}}{b-a}\; \;

This is an incorrect option.

Option 4)

\; \; \frac{b^{n+1}-a^{n+1}}{b-a}\; \; \;

This is the correct option.

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