geometric series

Question

The first 3 terms of a geometric expression are x, 10 − x ,and  2x + 1 where x > 0.

Find the value of the common ratio, r.

Answer 1

Let r = the common ratio

 

To find the common ratio of a geometric sequence, you divide any term by the previous term

Dividing the 2nd term by the 1st,

 

      r  = (10 - x)/x

 

Dividing the 3rd term, by the 2nd,

 

      r = (2x + 1)/(10 - x)

 

We can set these 2 equations equal to each other .... r = r

 

      (10 - x)/x  = (2x + 1)/(10 - x)

 

Cross multiply

 

      (10 - x)(10 - x) = x(2x + 1)

      100 - 10x - 10x + x² = 2x² + x

      x² - 20x + 100 = 2x² + x

      0 = 2x² - x² + 20x  + x - 100

      0 = x² + 21x - 100  

      0 = (x - 4)(x + 25)

 

Using zero product property,

 

      x - 4 = 0 ... x = 4

 

or

      x + 25 = 0 ... x = -25 ... discard this because it was given that x > 0

 

Therefore, x = 4

 

From an equation earlier,

 

   r = (10 - x)x

     = (10 - 4)/4

     = 6/4

     = 3/2

 

The common ratio r is 3/2

 

So the 1st 3 terms are

 

     = x, 10 - x, 2x + 1 

     = 4, 10 - 4, 2(4) + 1

     = 4,  6, 9

   

 

Answer 2

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