Geometric progression

Question

The second term of a convergent geometric series is 24 and the sum of the first three terms is 126. Find the common ratio of the series.

Answer 1

The geometric series has the form a + ar + ar² + ...

 

We have:  ar = 24 and a + ar + ar² = 126

 

From the first equation, a = 24/r

 

Replace a by 24/r in the second equation:  24/r + 24 + 24r = 126

 

24/r + 24r = 102

 

Multiply by r:  24 + 24r² = 102r

 

24r² - 102r + 24 = 0

 

Divide both sides by 6:  4r² - 17r + 4 = 0

 

                                          (4r - 1)(r - 4) = 0

 

                                          r = 1/4 or r =4

 

Since the series is convergent, -1 < r < 1, so r can't be 4.

 

Therefore, r = 1/4