A) 0
B) \[{{3}^{n}}\]
C) \[{{5}^{n}}\]
D) None of these
Correct Answer: A
Solution :
Let\[S\]denotes the sum of the series. The general term of the given series is \[{{T}_{r}}={{(-1)}^{r}}{{(3+5r)}^{n}}{{C}_{r}}\](\[nth\]term of AP) \[\therefore \] \[S=\sum\limits_{r=0}^{n}{{{(-1)}^{r}}}{{(3+5r)}^{n}}{{C}_{r}}\] \[=3\sum\limits_{r=0}^{n}{{{(-1)}^{r}}^{n}{{C}_{r}}}+5\sum\limits_{r=0}^{n}{{{(-1)}^{r}}^{n}{{C}_{r}}}\] \[=3({{C}_{0}}-{{C}_{1}}+{{C}_{2}}-{{C}_{3}}+{{C}_{4}}-....\] \[+{{(-1)}^{n}}\cdot {{C}_{n}})+5(-{{C}_{1}}+2{{C}_{2}}-3{{C}_{3}}\] \[+4{{C}_{4}}-...+{{(-1)}^{n}}\cdot n\cdot {{C}_{n}})\] \[\Rightarrow \] \[S=0+0=0\]
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