A) \[{{A}^{2}}-{{B}^{2}}\]
B) \[{{A}^{2}}+{{B}^{2}}\]
C) 4AB
D) None of these
Correct Answer: A
Solution :
[a] \[{{(x+a)}^{n}}={{\,}^{n}}{{C}_{0}}{{x}^{n}}+{{\,}^{n}}{{C}_{1}}{{x}^{n-1}}a+{{\,}^{n}}{{C}_{2}}{{x}^{n-2}}{{a}^{2}}\] \[+{{\,}^{n}}{{C}_{3}}{{x}^{n-3}}{{a}^{3}}+{{\,}^{n}}{{C}_{4}}{{x}^{n-4}}{{a}^{4}}+....\] \[={{(}^{n}}{{C}_{0}}{{x}^{n}}+{{\,}^{n}}{{C}_{2}}{{x}^{n-2}}{{a}^{2}}+{{\,}^{n}}{{C}_{4}}{{x}^{n-4}}{{a}^{4}}+....)+\] \[{{(}^{n}}{{C}_{1}}{{x}^{n-1}}a+{{\,}^{n}}{{C}_{3}}{{x}^{n-3}}{{a}^{3}}+{{\,}^{n}}{{C}_{5}}{{x}^{n-5}}{{a}^{5}})+....\] \[=A+B....(1)\] Similarly, \[{{(x-a)}^{n}}=A-B....(2)\] Multiplying eqns. (1) and (2), we get \[{{({{x}^{2}}-{{a}^{2}})}^{n}}={{A}^{2}}-{{B}^{2}}\]
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