Final answer:
The common ratios of all such GPs when the sum and product of four positive consecutive terms are given can be found by setting up equations using the properties of geometric progressions. By finding the values of the terms and the common ratio, we solve for the sum of the common ratios.
Step-by-step explanation:
When dealing with a geometric progression (GP), and given the sum and product of four positive consecutive terms as 126 and 1296, respectively, we can label the four terms as a/r³, a/r, ar, and ar³ (where a is the second term, and r is the common ratio). The sum of these terms equals 126, which can be written as:
a/r³ + a/r + ar + ar³ = 126
By multiplying everything by r³, we get:
a + ar² + ar´ + ar¶ = 126r³
The product of these terms is 1296, therefore:
(a/r³) * (a/r) * (ar) * (ar³) = 1296
Simplifying this, we get:
a^4 = 1296
So a = −6.
We can substitute a into the sum equation and solve for r³. Once r is found, we can fit it into the equation for the sum of the common ratios, which is r³ + r + 1/r + 1/r³.