If the expansion in powers of x of the function 11-ax1-bx is a0-a1x-a2x2-a3x3- then an is

The correct option is C a^n+1 b^n+1a b (1 ax)^ 2(1 bx)^ 1=(1+ax+a^2x^2......)(1+bx+b^2x^2.......) a_n= coefficient of x^n in (1 ac)^ ...

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Standard XII

Mathematics

Middle Terms

If the expans...

Question

If the expansion in powers of $x$ of the function $\frac{1}{(1 - a x) (1 - b x)}$ is $a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . .,$ then $a_{n}$ is

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

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

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

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

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Solution

The correct option is C $\frac{a^{n + 1} - b^{n + 1}}{a - b}$
$(1 - a x)^{- 2} (1 - b x)^{- 1} = (1 + a x + a^{2} x^{2} . . . . . .) (1 + b x + b^{2} x^{2} . . . . . . .)$
$a_{n} =$ coefficient of $x^{n}$ in $(1 - a c)^{- 1} (1 - b x)^{- 1}$
$= a^{o} b^{n} + a b^{n - 1} + . . . . . . . . . . . . . . . a^{n} b^{o}$
$= a^{o} b^{n} (1 + \frac{a}{b} + {(\frac{a}{b})}^{2} + . . . . . . . . . . .) = a^{o} b^{n} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{{(\frac{a}{b})}^{n + 1} - 1}{\frac{a}{b} - 1} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
[Sum of $n^{t h}$ term of a $G P$ with common ratio $\frac{a}{b}$ ]
$= \frac{b^{n} (a^{n + 1} - b^{n + 1)}}{(a - b)} . \frac{b}{b^{n + 1}} = \frac{a^{n + 1} - b^{n + 1}}{a - b}$

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If the expansion in powers of x of the function 1(1−ax)(1−bx) is a0+a1x+a2x2+a3x3+...., then an is

If the expansion in powers of $x$ of the function $\frac{1}{(1 - a x) (1 - b x)}$ is $a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . .,$ then $a_{n}$ is