Question
Let \( a,b,c,d,e\) are non-zero and distinct positive real numbers. If \( a,b,c\) are in \(AP\), \(b,c,d\) are in \(GP\). and \( c,d,e\) are in \(HP\), then \( a,c,e\) are in :
(a) \(AP\)
(b) \(GP\)
(c) \(HP\)
(d) Nothing can be said
(a) \(AP\)
(b) \(GP\)
(c) \(HP\)
(d) Nothing can be said
Answer
If \( a,b,c\) A.P \( \Rightarrow b=\frac{a+c}{2}\)
if \( c,d,e \) H.P. \( \Rightarrow d=\frac{2ec}{e+c}\)
if \( b,c,d \) G.P. \( \Rightarrow {c}^{2}=bd\)
\( {c}^{2}=\left(\frac{a+c}{2}\right)\left(\frac{2ec}{e+c}\right)\)
\( \Rightarrow c^{2} =\frac{(a+c) e c}{e+c} \\ \Rightarrow c=\frac{(a+c) e}{e+c} \\ \Rightarrow c(e+c)=(a+c) e \\ \Rightarrow c e+c^{2}=a e+c e \\ \Rightarrow c^{2}=a e\)
Hence, the correct option is (b).
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