Given a, b, c are in A.P., b, c, d are in G.P., and c, d, e are in H.P. If a = 2 and e = 18, then the sum of all possible value of c is _________.

a, b, c are in A.P. $\Rightarrow \mathrm{b}=\frac{\mathrm{a}+\mathrm{c}}{2}$         (1)

b, c, d are in G.P.

$\Rightarrow {\mathrm{c}}^2=\mathrm{bd}$                                    (2)

and c, d, e are in H.P.

$\Rightarrow \mathrm{d}=\frac{2\mathrm{ce}}{\mathrm{c}+\mathrm{e}}$                                (3)

Now c² = bd

or ${\mathrm{c}}^2=\left(\frac{\mathrm{a}+\mathrm{c}}{2}\right)\left(\frac{2\mathrm{ce}}{\mathrm{c}+\mathrm{e}}\right)$            [using (1) and (3)]

$\therefore {\mathrm{c}}^2+\mathrm{ce}=\mathrm{ae}+\mathrm{ce}$

or c² = ae

Now given a = 2 and e = 18

$\therefore {\mathrm{c}}^2=\mathrm{ae}$

or ${\mathrm{c}}^2=2\times 18=36$

$\Rightarrow \mathrm{c}=6\text{ or }-6$