A) \[\frac{{{3}^{n}}+{{(-1)}^{n}}}{2}\]
B) \[\frac{{{3}^{n}}-{{(-1)}^{n}}}{2}\]
C) \[\frac{{{3}^{n}}+1}{2}\]
D) \[\frac{{{3}^{n}}-1}{2}\]
Correct Answer: B
Solution :
\[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+{{C}_{3}}{{x}^{3}}+.....+{{C}_{n}}{{x}^{n}}\] \[{{(1-x)}^{n}}={{C}_{0}}-{{C}_{1}}x+{{C}_{2}}{{x}^{2}}-{{C}_{3}}{{x}^{3}}+.....+{{(-1)}^{n}}{{C}_{n}}{{x}^{n}}\]\[[{{(1+x)}^{n}}-{{(1-x)}^{n}}]=2\,[{{C}_{1}}x+{{C}_{3}}{{x}^{3}}+{{C}_{5}}{{x}^{5}}+...]\] \[\frac{1}{2}[{{(1+x)}^{n}}-{{(1-x)}^{n}}]={{C}_{1}}x+{{C}_{3}}{{x}^{3}}+{{C}_{5}}{{x}^{5}}+.......\] Put x = 2, \[2.{{C}_{1}}+{{2}^{3}}.{{C}_{3}}+{{2}^{5}}.{{C}_{5}}+.....\,=\frac{{{3}^{n}}-{{(-1)}^{n}}}{2}\]
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