Introduction to Geometric Series and the Common Ratio

Geometric series are sequences of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Understanding the common ratio is crucial for solving various mathematical problems related to these series.

Problem Statement: Sum of Terms in a Geometric Series

Suppose the sum of the first four terms of a positive geometric series is five times the sum of the first two terms. We need to find the common ratio, denoted as r.

Breaking Down the Series

Let the first term of the geometric series be denoted by a. The series’ terms are as follows:

First term: a Second term: ar Third term: ar^2 Fourth term: ar^3

Sum of the First Four Terms

The sum of the first four terms, S4, can be expressed as:

[ S_4 a ar ar^2 ar^3 ]

Factoring out a, we get:

[ S_4 a(1 r r^2 r^3) ]

Sum of the First Two Terms

The sum of the first two terms, S2, is:

[ S_2 a ar ]

Factoring out a, we get:

[ S_2 a(1 r) ]

Establishing the Relationship

Given that the sum of the first four terms is five times the sum of the first two terms:

[ S_4 5S_2 ]

Substituting the expressions for S4 and S2, we obtain:

[ a(1 r r^2 r^3) 5a(1 r) ]

Assuming a ≠ 0

Since a is not equal to zero, we can divide both sides by a (assuming a non-zero a):

[ 1 r r^2 r^3 5(1 r) ]

Simplifying the right side:

[ 1 r r^2 r^3 5 5r ]

Rearranging the equation gives:

[ r^3 r^2 - 4r - 4 0 ]

Solving the Cubic Equation

We use the Rational Root Theorem to test for possible rational roots. Testing r 2:

[ 2^3 2^2 - 4(2) - 4 0 ]

Simplifying this:

[ 8 4 - 8 - 4 0 ]

Thus, r 2 is a root. We can factor the cubic polynomial using synthetic division:

 2 | 1  1  -4  -4   |    2  6  4   ------------     1  3  2  0

This gives:

[ (r - 2)(r^2 3r 2) 0 ]

Factoring r^2 3r 2:

[ r^2 3r 2 (r 1)(r 2) ]

Thus, the complete factorization is:

[ (r - 2)(r 1)(r 2) 0 ]

The solutions to this equation are:

[ r 2, quad r -1, quad r -2 ]

Since we are looking for the common ratio of a positive geometric series, the only valid solution is:

[ boxed{2} ]

Conclusion

The common ratio of the geometric series, given the conditions, is 2.