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) 36
b) 24
c) 18
d) 12

• (log a + log b + log c)/log 6 = 6
  (log abc )/log 6 = 6
  log6 abc=6
  abc = 6^6
  from gp series a/r * a * ar =6^6
  a^3= 6^6
  a=36
  so, a= 36/r, b=36, c=36r
  c-b = 36r-36
  =36(r-1)
  ans is 36
• 8 years agoHelpfull: Yes(14) No(3)
• let a=(a/r),b=a,c=ar
  (log a + log b + log c)/log 6 = 6
  (log abc )/log 6 = 6
  log6 abc=6
  abc = 6^6
  So a/r * a * ar =6^6
  a=36.
  a = (36/r), b = 36, c = 36r.
  c-b=36(r-1)
  but it is given that r is any number.
  for r=1 ,it gives 0, which not satisfied the condition.
  Finally for r = 4/3, then c-b=12
  
  
• 8 years agoHelpfull: Yes(10) No(3)
• given (log a +log b+ log c)/log 6=6
  then 1/6(log(abc))=log6
  log(abc)^1/6=log 6
  eliminating log both side
  (abc)^1/6=6
  abc=6^6--(1)
  now
  assume three term k/r,k,kr which is a,b,c
  substitute in eq 1
  k/r*k*kr=6^(3*2)
  k^3=6^3*6^2
  K=6^(3*1/3)*6^2
  i.e k=6^2=36=b(given b is gp term)
  so a=k/r=36/r & c=36r
  finally c-b=36r-36=36(r-1)
  (c-b)/(r-1)=36 since r any real no so it independent of r for any value ie c-b=36
  so ans is 36
  
  
• 8 years agoHelpfull: Yes(3) No(1)
• the answer is 12 because he is asking minimum value of (c-b)
• 8 years agoHelpfull: Yes(3) No(0)
• a^3= 6^6
  a=36
  How are you guys writing this?Please explain
• 8 years agoHelpfull: Yes(2) No(0)
• k log m=log m^k
  that why they write as 6 log 6 as 36
• 8 years agoHelpfull: Yes(2) No(0)
• (log a+log b+log c)/log 6=6
  since k log m = log m^k & (log a+log b+log c) =log abc by log rules
  log abc=log 6^6
  if we cancel out log on both sides
  abc=36
  a,b,c are the geometric progression so a*b*c=(a*ar*ar^2)
  so (a^3)*(r^3)=6^3
  so ar=36
  a=36/r
  b=a*r=(36/r)/r=36 and c=36r
  c-b=36(r-1)=36(when r=2)
• 8 years agoHelpfull: Yes(1) No(0)
• Here the answer Will be 12 because r can be any real no.
  (Only for positive integer r, the answer will be 36.)
• 8 years agoHelpfull: Yes(1) No(0)
• what is the answer for this question..Is it 12 or 36......?
• 8 years agoHelpfull: Yes(0) No(0)
• (log a + log b + log c)/log 6 = 6
  (log abc )/log 6 = 6
  log6 abc=6
  abc = 6^6
  from gp series a/r * a * ar =6^6
  a^3= 6^6
  a=36
  so, a= 36/r, b=36, c=36r
  c-b = 36r-36
  =36(r-1)
  ans is 36
• 8 years agoHelpfull: Yes(0) No(0)
• a^3 =6^6
  
  (a^(1/2))^(2*3) =6^6
  
  ( sqrt(a) )^6 =6^6
  
  as powers on both sides are equal equate the bases
  
  ( sqrt(a) ) =6
  
  so a= 6^2 = 36
• 8 years agoHelpfull: Yes(0) No(0)
• (log a + log b + log c)/log 6 = 6
  (log abc )/log 6 = 6
  log6 abc=6
  abc = 6^6
  from gp series a/r * a * ar =6^6
  a^3= 6^6
  a=36
  so, a= 36/r, b=36, c=36r
  c-b = 36r-36
  =36(r-1)
  ans is 36
• 8 years agoHelpfull: Yes(0) No(0)