• # question_answer If $a,\,\,\,b,\,\,\,c$ are in $A.P.,\,\,\,b,\,\,\,c,\,\,\,d$ are in $GP.$ and $c,\,\,\,d,\,\,\,e$ are in $H.P.$ then $a,\,\,\,c,\,\,\,e$ are in A) $A.P.$                               B) $GP.$C) $H.P.$               D)  None

$a,\,\,b,\,\,c$in$A.P.$$\Rightarrow a+c=2b;$  $b,\,\,c,\,\,d$in$G.P.$ $\Rightarrow$  $bd={{c}^{2}};$   $c,\,\,d,\,\,e$in$H.P.$$\Rightarrow d=\frac{ce}{c+e}$ $\therefore$$\frac{a+c}{2}\times \frac{2ce}{c+e}={{c}^{2}}\Rightarrow (a+c)e=(c+e)c\Rightarrow {{c}^{2}}=ae$Therefore, in$G.P.$