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Step-by-step explanation:
To find the common ratio (r) of the geometric series, we can use the given information about the sums of the terms.
1. Let's denote the first term of the series as "a" and the common ratio as "r". The sum of the first two terms can be written as:
a + ar = 10
2. The sum of the first four terms can be written as:
a + ar + ar^2 + ar^3 = 30
3. Dividing the equation for the sum of the first four terms by the equation for the sum of the first two terms, we get:
(a + ar + ar^2 + ar^3) / (a + ar) = 30 / 10
4. Simplifying the equation, we have:
1 + r + r^2 + r^3 = 3
5. Rearranging the terms, we have:
r^3 + r^2 + r - 2 = 0
6. We can see that the equation is a cubic equation. By observation, we notice that r = 1 satisfies the equation. Therefore, we can factor out (r - 1) from the equation:
(r - 1)(r^2 + 2r + 2) = 0
7. To solve for the remaining quadratic equation, we can use the quadratic formula:
r = (-2 ± √(2^2 - 4(1)(2))) / (2(1))
8. Simplifying the expression, we get:
r = (-2 ± √(4 - 8)) / 2
r = (-2 ± √(-4)) / 2
9. Since we cannot take the square root of a negative number in the real number system, there are no real solutions for the quadratic equation.
10. Therefore, the only valid solution for the common ratio r is r = 1.
