A) 1
B) 0
C) 2
D) 3
Correct Answer: A
Solution :
Let \[a\] and \[b\] two numbers respectively. Sum of \[n\] A.M.'s \[=n\times \]single A.M. \[\Rightarrow \]\[{{A}_{1}}+{{A}_{2}}=2\times \left( \frac{a+b}{2} \right)=a+b\] Product of \[n\] G.M.'s = (Single G.M.)n \[\Rightarrow \] \[{{G}_{1}}.{{G}_{2}}={{(\sqrt{ab})}^{2}}=ab\] \[\frac{1}{a},\ \frac{1}{{{H}_{1}}},\ \frac{1}{{{H}_{2}}},\ \frac{1}{b}\] are in A.P. \[\Rightarrow \] \[\frac{1}{{{H}_{1}}}+\frac{1}{{{H}_{2}}}=\frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab}\]\[\Rightarrow \]\[\frac{{{H}_{1}}{{H}_{2}}}{{{H}_{1}}+{{H}_{2}}}=\frac{{{G}_{1}}{{G}_{2}}}{{{A}_{1}}+{{A}_{2}}}\] \[\Rightarrow \] \[\frac{{{G}_{1}}{{G}_{2}}}{{{H}_{1}}{{H}_{2}}}\times \frac{{{H}_{1}}+{{H}_{2}}}{{{A}_{1}}+{{A}_{2}}}=1\]
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