A) \[\frac{{{3}^{n}}+1}{2}\]
B) \[\frac{{{3}^{n}}-1}{2}\]
C) \[\frac{{{3}^{n-1}}+1}{2}\]
D) \[\frac{{{3}^{n-1}}-1}{2}\]
Correct Answer: A
Solution :
We have, \[{{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+{{a}_{3}}{{x}^{3}}+{{a}_{4}}{{x}^{4}}+...+{{a}_{2n}}{{x}^{2n}}\] \[={{(1-x+{{x}^{2}})}^{n}}\] On putting \[x=1\] and\[-1\], respectively we get \[({{a}_{0}}+{{a}_{2}}+{{a}_{4}}+...)+({{a}_{1}}+{{a}_{3}}+{{a}_{5}}+...)=1\] ... (i) and\[({{a}_{0}}+{{a}_{2}}+{{a}_{4}}+...)\] \[-({{a}_{1}}+{{a}_{3}}+{{a}_{5}}+...)={{3}^{n}}\] ... (ii) On adding Eqs. (i) and (ii), we get \[2({{a}_{0}}+{{a}_{2}}+{{a}_{4}}+...)={{3}^{n}}+1\] \[{{a}_{0}}+{{a}_{2}}+{{a}_{4}}+...=\frac{{{3}^{n}}+1}{2}\]
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