The correct option is D 480
(i) Sum=a+b+c=25 ........(1)
(ii) The terms 2,a,b are in A.P
⇒2a=b+2 .........(2)
(iii) The terms b,c,18 are in G.P
⇒c2=18b ....(3)
From (1) and (2) a+b+c=25
⇒a=25−b−c
(on transposition from L.H.S to R.H.S)
⇒b+22=25−b−c (from (2))
⇒b+2=50−2b−2c (Multiplying both sides by 2)
⇒b+2b=48−2c or 3b=48−2c
From(3) substituting for b we get
2c+3c218=48
⇒2c+c26=48
⇒12c+c2−288=0 (Multiplying by 6)
⇒c2+12c−288=0 (Rearranging)
⇒(c−12)(c+24)=0 (on factorizing the above equation)
Hence. c=12,−24
c=12 is valid (∵a,b,c) lie between 2 and 18
∴c=12⇒c2=18b (from (3))
Substituting c=12 in c2=18b
we get b=14418=8
Substituting the values of b=8,c=12 in a+b+c=25 we get a=25−20=5
∴a=5,b=8,c=12
abc=5×8×12=480