Question

If $a,b,c$ be in A.P, $b,c,d$ are in G.P and $c,d,e$ are in H.P, then $a,c,e$ are in ...........................

• A. H.P
• B. A.P
• C. G.P
• D. both H.P and G.P

Answer 1

The correct option is C G.P

Since $a,b,c$ are in $A.P.$

$\therefore b=\frac{a+c}{2}$ ...(1)

$\because b,c,d$ are in $G.P.$

$\therefore c^2=bd$ ...(2)

Also, $\because c,d,e$ are in $H.P.$

$\therefore d=\frac{2ce}{c+e}$ ...(3)

Substituting the values of $b$ and $d$ from (1) and (3) respectively in (2), we get

$c^2=\frac{a+2}{2}\times \frac{2ce}{c+e}\Rightarrow c=\frac{e(a+c)}{c+e}$

$\Rightarrow c^2+ce=ae+ce$

$\Rightarrow c^2=ae$

$\Rightarrow a,c,e$ are in $G.P.$