Question

If a,b,c,d,e be five numbers such that a,b,c are in A.P., b,c,d are in G.P., and c,d,e are in H.P. prove that
If a = 2 and e = 18, such that find all positive values of b,c and d.

Answer 1

a = 2 and e = 18
We have $c^2=ae$
$or{c}^2=36\therefore =c=6,-6$
2b = a + c, by(1)
or 2b = 2 + 6 or 2 - 6
$\therefore$ b = 4, -2
$c^2=bd\left(2\right)$
36 = 4d or (-2)d
$\therefore$ d = 9, -18
Hence b,c,d are 4,6,9 or -2,-6,-18
$\therefore$ The five numbers a,b,c,d,e are
2,4,6,9,18 or 2,-2,-6,-18,18