Three numbers X, Y, and Z are in the ratio 2:7:8. If 12 is subtracted from Y, then three numbers form a geometric sequence (in the order X, Y-12, Z). Find X, Y, and Z.

A geometric sequence is a sequence in which there is a common ratio between any two consecutive terms. In this case if X:Y:Z are in the ratio of 2:7:8 the multiplying by a constant k, we have X=2k, Y = 7k and Z=8k.

Then if X, Y-12, Z form a Geometric sequence, it means X/Y-12=Y-12/Z which is the same as 2k/7k-12=7k-12/8k if we cross multply, we get

16k² = 49k²-168k + 144

33k²-168k+144 = 0 solving for k

k = 4 or 1.091 if we take the whole number to find the values of X, Y and z,

X = 8, Y = 28 and Z=32