product of 11th and 15 th term of GP is 3.what would be the 5th root of the product of 25 terms of GP.

• nth term of an gp is ar^n-1
  
  so 11th term is ar^10 and 15th term is a^24
  
  now ar^10*ar^14=3 i.e. a^2r^24=3.
  
  now,
  product of 25th term=a*ar*ar^2....*ar^24.
  so, a^25r^(1+2+3...+24)
  so,
  a^25r^300.
  
  because 1+2+3..+24=24*(1+24)/2=300.
  
  now, 5th square root of product of 25th term is a^5*r^60 ie. a^2r^24*a^2r^24*ar^12 i.e 3*3*sq_root(3) = 9*sq.root(3).
• 9 years agoHelpfull: Yes(26) No(1)
• Let the first 25 terms of GP be ar^-12 ,ar^-11 ,ar^-10,............ ,a,ar,.............. ,ar^11 ,ar^12
  Therefore Product of 25 terms of GP is a^25 .
  5th root of product of first 25 terms is (a^25)^1/5 = a^5.
  11th term in above series is ar^-2
  15th term is ar^2
  So product of 11th and 15th terms is a^2 =3
  Therfore answer is a^5 = a^2 * a^2 *a
  =3 * 3* sq.rt(3)
  =9 {sq.rt (3)}
• 9 years agoHelpfull: Yes(8) No(0)
• let first term be a and common difference be d.
  so ar^10*ar^14=3 now as 3 is either equal to 1*3 or3*1 ,hence ar^110=1 and ar^14=3
  dividing ar^14/ar^10=r^4=3 i.e r=3 ^1/4 a=1/r^10=1/3^10/4=1/3^5/2
  now a*ar*ar^2..........ar^24=product of first 25 terms=a^25*r^1+2+3+4......+24=a^25*r^300=(1/3^5/2)^25*3^300/4
• 9 years agoHelpfull: Yes(1) No(0)
• let the terms be a , ar, ar^2.......................................
  now the product of 11th and 15th term will be (ar^10*ar14) = a^2r^24 = 3
  now product of 25 terms (a*ar*ar^2.............................*ar^24)=a^25*r^(0+1+2+3.................................+24)
  now(0+1+2.........24)= 300
  therefore it becomes a^25*r^300
  
  now its fifth root=( a^25*r^60)^1/5= a^5*r^60
  now (a^5*r^12)^2=a^10*r^120
  
  now,(a^2*r^24)^5=3^5=a^10*r^120
  therefore sqrt(a^10*r^120)= a^5*r^60=sqrt(3^5)
  =9sqrt(3) Ans
  
• 9 years agoHelpfull: Yes(0) No(0)
• a^2*r^24=3 a*r^12=3
  product of 25 terms is a^5*r^300
  so 5th root is 3^0.5
• 9 years agoHelpfull: Yes(0) No(0)
• ANS 9*sqroot(3)
  formula of GP is (a.r^n-1)
  here (11th * 15th)term=3
  ar^10*ar^14=3
  a^2r^24=3--------------------------------------------1
  and here 25 term not 25th term(25 term and 25th term totaly different)
  a*ar*ar^2*ar^3*...............*ar^24
  a^25 r^(1+2+3+4+...................+24)
  by formula of AP we no
  1+2+3+4+.....................+n=n(n+1)/2
  so
  a^25 r^(1+2+3+4+...................+24)
  (1+2+3+4+...................+24)
  here n=24
  24(24+1)/2=12*25=300
  a^25 r^300
  but in qustion 5th root of the product of 25 terms of GP
  a^5r^60
  (by this equation a^2r^24=3--------------------------------------------1)
  a^5r^60=a^2r^24 a^2r^24 a r^12(we can write)
  3 3 sqr root of 3
  
  ANS 9*sqroot(3)
  
• 9 years agoHelpfull: Yes(0) No(0)