The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is 27 19 . Then the common ratio of this series is:

Step-by-step solution

Let the terms of infinite series are a, a r, a r², a r³, So, a 1-r =3 Since, sum of cubes of its terms is 27 19 that is sum of a³ , a³ r³, is 27 19 So, a³ 1-r³ = 27 19 a 1-r a² (1+r²+r ) = 27 19 9 (1+r²-2 r ) 3 1+r²+r = 27 19 6 r²-13 r+6=0 (3 r-2)(2 r-3)=0 r= 2 3 , or 3 2 As |r| < 1 So, r= 2 3