• # question_answer Let ${{(1+{{x}^{2}})}^{2}}\,{{(1+x)}^{n}}={{A}_{0}}+{{A}_{1}}x+{{A}_{2}}{{x}^{2}}+.....$ If ${{A}_{0}},$${{A}_{1}},$${{A}_{2}}$ are in A. P. then the value of n is: A) 2 B) 4 C) 5 D) 7

 ${{(1+{{x}^{2}})}^{2}}\,\,{{(1+x)}^{n}}=(1+2{{x}^{2}}+{{x}^{4}})\,\,{{(}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}x+$$^{n}{{C}_{2}}{{x}^{2}}+.....)$${{=}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}x+{{(}^{n}}{{C}_{2}}+2{{\cdot }^{n}}{{C}_{0}})\,{{x}^{2}}+.....$ Hence ${{A}_{0}}=1\,\,;~\,\,{{A}_{1}}={}^{n}{{C}_{1}}\,\,;\,\,{{A}_{2}}={}^{n}{{C}_{2}}\,+2$  which are in A. P.