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If \( a_{0}, a_{1}, a_{2}, \ldots a_{n} \) is an arithmetic progression of positive integers, then to evaluate
\( \frac{C_{0}}{a_{0}} \pm \frac{C_{1}}{a_{1}}+\frac{C_{2}}{a_{2}} \pm \cdots \) upto \( (n+1) \) terms, we integrate both
the sides of
\[
\sum_{r=0}^{n} C_{r} x^{r}=(1+x)^{n}
\]
after multiplying by \( x^{r} \) for an appropriate value of \( r \).
\( 3 C_{0}+\frac{3^{2}}{2} C_{1}+\frac{3^{3}}{3} C_{2}+\cdots \) upto \( (n+1) \) terms equals
(a) \( \frac{1}{n+1}\left(4^{n+1}-1\right) \)
(b) \( \frac{n}{n+3}\left(4^{n+1}+1\right) \)
(c) \( \frac{1}{n+2}\left(4^{n+1}-1\right) \)
(d) \( \frac{1}{n+1}\left(4^{n+1}+1\right) \)
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