Let the median number be m and let the ratio be r. Then the members of the progression are m/r, m, m*r.
Their sum is mr + m + m*r = m*(1/r + 1 + r).
The reciprocals of the progression are 1/(m*r), 1/m, r/m.
Their sum is 1/(m*r) + 1/m + r/m = (1/m)*(1/r + 1 + r).
From the given values m*(1/r + 1 + r)=50 and (1/m)*(1/r + 1 + r)=4.5.
Create a ratios equation be dividing the first equation by the second:
[m*(1/r + 1 + r)]/[(1/m)*(1/r + 1 + r) = 50/4.5
This simplifies to m^2 = 100/9.
Then m = 10/3 or -10/3. If m=10/3 then (10/3)*(1/r + 1 + r) = 50, which simplifies to 1/r + r = 14. This implies r = 7+4*sqrt(3) or 7-4*sqrt(3). These two answers are actually the same geometric sequence in ascending and descending order.
If m=-10/3 then then (-10/3)*(1/r + 1 + r) = 50, which simplifies to 1/r + r = -16. This implies r = -8+3*sqrt(7) or -8-3*sqrt(7). Like the previous case these two answers describe the same sequence.
Then the two distinct geometric progressions are
{(70-40*sqrt(3))/3, 10/3, (70+40*sqrt(3))/3} and
{(80-30*sqrt(7))/3, -10/3, (80+30*sqrt(7))/3}.
Solution
