The three numbers a, b, c between 2 and 18 are such  that their sum is 25; the numbers 2, a, b are consecutive terms of an A.P. and the numbers b, c, 18 are  consecutive terms of a G.P. The three numbers are

we have a+b+c=25       (1)∵2,a,b are in A.P., ∴ 2a=2+b   (2)∵b,c18 are in G.P., ∴ c2=18b    (3)From Eq. (1) and (2),   3b+2c=48; or 3b=48−2c∴ From Eq. (3),   c2=6(48−2c)=288−12cor  c2+12c−288=0or  c2+24c−12c−288=0 (∵a=b+12)or  18b=−72∴b=−4,From Eq. (1),  a=−4+12=8 Hence, the four numbers are 8, –4, 2 and 8.