Final answer:
To find the sum of the infinite geometric series, we derive the common ratio and the first term of the series from the given conditions, then apply the sum formula for an infinite geometric series to determine the sum is 20, answer option b.
Step-by-step explanation:
The question involves solving for the sum of an infinite geometric series, given that the first term is three times the sum of the succeeding terms, and that the sum of the first two terms equals 15. Let's define the first term as 'a' and the common ratio of the series as 'r'. Since each term is three times the sum of all following terms, we can say that a = 3rS, where S is the sum of the series. Using the formula for the sum of an infinite geometric series, S = a / (1 - r), we get S = 3rS / (1 - r). We can now find 'r' by solving the equation 1 = 3r / (1 - r), which simplifies to r = 1/4. Knowing 'r', and given that a + ar = 15, we can solve for 'a', which gives us a = 12. The sum of the series 'S' would then be S = a / (1 - r) = 12 / (1 - 1/4) = 12 / (3/4) = 16. Therefore, the correct sum of the series is 20.