AP EAMCET · Maths · Binomial Theorem
If \(\frac{(1-p x)^{-1}}{(1-q x)}=a_0+a_1 x+a_2 x^2+a_3 x^3+\ldots\), then \(a_n=\)
- A \(\frac{p^{n+1}-q^{n+1}}{q-p}\)
- B \(\frac{p^{n+1}-q^{n+1}}{p-q}\)
- C \(\frac{p^n-q^n}{q-p}\)
- D \(\frac{p^n-q^n}{p-q}\)
Answer & Solution
Correct Answer
(B) \(\frac{p^{n+1}-q^{n+1}}{p-q}\)
Step-by-step Solution
Detailed explanation
Since, \(\begin{aligned} & \frac{(1-p x)^{-1}}{(1-q x)}=a_0+a_1 x+a_2 x^2+a_3 x^3+\ldots \\ & \because(1-p x)^{-1}=1+p x+p^2 x^2+p^3 x^3+\ldots+p^n x^n+\ldots \end{aligned}\) and \((1-q x)^{-1}=1+q x+q^2 x^2+q^3 x^3+\ldots+q^n x^n+\ldots\) Now, coefficient of \(x^n\) in the…
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