If a, b and c are forming increasing terms of G.P. , r is the common ratio then find the minimum value of (c-b), given that (log a+log b+log c)/log 6 =6.Note that r can be any real no.

• a,b,c are in GP so let the first term of gp is a with common ration r.
  then b=ar
  c=ar^2
  Now (loga+logb+logc)/log6=6
  logabc (base 6) =6
  abc=6^6
  put the value of a,b,c in gp format
  a.ar.ar^2=6^6
  a^3r^3=6^6
  ar=6^2
  ar=36;
  that is b=36.
  Now we've been asked minimum value of (c-b)
  there was 4 option
  A).36 b).24 c). 18 D). 12
  lets take c-b =18
  you will get c= b+18 =>54
  now b=ar=36
  r=54/36
  so r =3/2
  then b=ar=36
  a.3/2=36
  a=24
  now we have gp as 24,36,54
  and c-b=18 ( i would recommend you to solve this by using option A, B also. Although option c is also not true)
  Now lets go with option 4
  lets take c-b=12
  c=12+b;
  c=48
  r=c/b;
  r=48/36;
  r=4/3;
  b=ar=36;
  a.4/3=36;
  a=27;
  Now the gp we have is the 27,36,48
  and the differnce between c and b is 12 that is minimum
  hence the option is D
  
  
  
• 8 years agoHelpfull: Yes(11) No(0)
• those whose ans are not belong to those options are wrong ..A).36 b).24 c). 18 D). 12 ...as those were the given options..
• 8 years agoHelpfull: Yes(1) No(2)
• 180
  after calculations ar=36
  for gp a,ar,ar^2
  6,6^2,6^3
  ar(r-1)=36(6-1)=180
• 4 years agoHelpfull: Yes(1) No(0)
• ans can be 6
• 8 years agoHelpfull: Yes(0) No(1)
• please explain!!!
• 8 years agoHelpfull: Yes(0) No(0)
• Answer is 30
• 8 years agoHelpfull: Yes(0) No(2)
• a,b,c are in G.P. so let the first term of G.P. =
  a
  r
  , and common ratio = r.
  Therefore, a =
  a
  r
  , b =
  a
  , c =
  a
  r
  
  Given,
  log
  a
  +
  log
  b
  +
  log
  c
  log
  6
  =
  6
  
  ⇒
  log
  a
  b
  c
  log
  6
  =
  6
  
  ⇒
  log
  6
  a
  b
  c
  =
  6
  ⇒
  a
  b
  c
  =
  6
  6
  
  put the value of a,b,c in gp format
  ⇒
  a
  r
  ×
  a
  ×
  a
  r
  =
  6
  6
  
  ⇒
  a
  3
  =
  6
  6
  ⇒
  a
  =
  36
  
  Now a =
  36
  r
  , b = 36, c = 36r.
  We have to find the minimum value of c - b = 36r - 36.
  r can be any number. So for r < 0, we get c - b negative.
  When r = 1, c - b = 0
  But none of the options are not representing it.
  From the given options, r = 4/3, then c = 48. So option d satisfies this.
• 5 years agoHelpfull: Yes(0) No(2)