A) \[n{{.2}^{n}}\]
B) \[~n{{.2}^{n+1}}\]
C) \[(n+1){{.2}^{n}}\]
D) \[n{{.2}^{n}}+1\]
Correct Answer: C
Solution :
[c] \[(x+{{\,}^{n}}{{C}_{0}})(x+3.{{\,}^{n}}{{C}_{1}})(x+5.{{\,}^{n}}{{C}_{2}})....\] |
\[(x+(2n+1).{{\,}^{n}}{{C}_{n}})\] |
\[={{x}^{n+1}}+{{x}^{n}}\{{{\,}^{n}}{{C}_{0}}+3.{{\,}^{n}}{{C}_{1}}+5.{{\,}^{n}}{{C}_{2}}+....\] |
\[+(2n+1).{{\,}^{n}}{{C}_{n}}\}+......\] |
Coeff. of \[{{x}^{n}}\] |
\[={{\,}^{n}}{{C}_{0}}+3.{{\,}^{n}}{{C}_{1}}+5.{{\,}^{n}}{{C}_{2}}+....+(2n+1).{{\,}^{n}}{{C}_{n}}\] |
\[=1+({{\,}^{n}}{{C}_{1}}+2.{{\,}^{n}}{{C}_{1}})+({{\,}^{n}}{{C}_{2}}+4.{{\,}^{n}}{{C}_{2}})+...\] |
\[+{{(}^{n}}{{C}_{n}}+2n.{{\,}^{n}}{{C}_{n}})\] |
\[=(1+{{\,}^{n}}{{C}_{1}}+...+{{\,}^{n}}{{C}_{n}})+2({{\,}^{n}}{{C}_{1}}+2{{\,}^{n}}{{C}_{2}}+...+n.{{\,}^{n}}{{C}_{n}})\] |
\[={{2}^{n}}+2\left[ n+2.\frac{n(n-1)}{2!}+3.\frac{n(n-1)(n-2)}{3!}+...+n.1 \right]\]\[={{2}^{n}}+2n[1+{{\,}^{n-1}}{{C}_{1}}+{{\,}^{n-1}}{{C}_{2}}+....+{{\,}^{n-1}}{{C}_{n-1}}]\] |
\[={{2}^{n}}+2n{{.2}^{n-1}}={{2}^{n}}(1+n)=(n+1){{.2}^{n}}\] |
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