geometric sequence.Determine the numbers a, b if 3 ,a ,b , 24 is a geometric sequence.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The consecutive terms of a geometric sequence have a common ratio.

We have the sequence given by 3 ,a ,b , 24

24/b = b/a = a/3

=> 24a = b^2 or a = b^2/24

substitute in b/a = a/3

=> 24b/b^2 = b^2/3*24

=> 24/b = b^2/24*3

=> b^3 = 24*24*3

=> b = 3^3*8*8

=> b = 3*4

=> b = 12

a = 144/24 = 6

The value of a = 6 and b = 12

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

a^2 = 3b (1)

b^2 = 24a (2)

We'll raise to square (1):

a^4 = 9b^2

We'll divide by 9 both sides:

b^2 = a^4/9 (3)

We'll substitute (3) in (2):

a^4/9 = 24a

We'll divide by a:

a^3/9 = 24

We'll cross multiply and we'll get:

a^3 = 24*9

a^3 = 2^3*3^3

a = 2*3

a = 6

For a = 6, we'll get b:

b^2 = 24a

b = sqrt 24*6

b = sqrt144

b = 12

Therefore, given a=6 and b=12, the terms of the geometric series, whose common ratio is r  =2, are: 3 , 6 , 12 , 24, ....

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