A) \[\frac{a}{{{2}^{n}}}\]
B) \[na\]
C) \[0\]
D) \[\frac{d}{{{2}^{n}}}\]
Correct Answer: C
Solution :
Required series can be written as \[a({{C}_{0}}-{{C}_{1}}+{{C}_{2}}+{{C}_{3}}+....)+d(-{{C}_{1}}+2{{C}_{2}}-{{C}_{3}}+....)\] ??(i) \[\because \,{{(1-x)}^{n}}={{C}_{0}}-{{C}_{1}}x+{{C}_{2}}{{x}^{2}}-{{C}_{3}}{{x}^{3}}.....+{{(-1)}^{n}}({{C}_{n}}{{x}^{n}})\] ?.(ii) Put \[x=1,\] we get \[{{C}_{0}}-{{C}_{1}}+{{C}_{2}}-{{C}_{3}}+....=0\] [A] Again differentiate equation (ii), we get \[-n{{(1-x)}^{n-1}}=-{{C}_{1}}+2{{C}_{2}}x-3{{C}_{3}}{{x}^{2}}\]\[+....+{{(-1)}^{n}}n{{C}_{n}}{{x}^{n-1}}\] .?(iii) Put \[x=1,\] we get \[-{{C}_{1}}+2{{C}_{2}}-3{{C}_{3}}+4{{C}_{4}}-.....=0\] ? From [A] and [B] Required series = 0.
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