Final answer:
The common ratio of the geometric series is 3 and the first term is 7. This is determined by solving the quadratic equation obtained from the relationship between the sums of the first three and six terms of the series.
Step-by-step explanation:
Let's denote the first term of the geometric series as a and the common ratio as r. The sum of the first three terms of a geometric series is given by S3 = a + ar + ar^2, which equals 77. Similarly, the sum of the first six terms is S6 = a + ar + ar^2 + ar^3 + ar^4 + ar^5, which equals 693.
To find the common ratio r, we can write the sum of the first six terms as the sum of the first three terms plus the next three terms in the series, which gives us S6 = S3 + ar^3 + ar^4 + ar^5. Replacing S3 with 77 and S6 with 693, we get 693 = 77 + ar^3(1 + r + r^2), leading to 616 = ar^3(1 + r + r^2).
Since ar^3 is the fourth term, it can be written as ar^3 = a + ar + ar^2 = 77, which simplifies our equation to 616 = 77(1 + r + r^2). Dividing both sides by 77 yields 8 = 1 + r + r^2, which simplifies to r^2 + r - 7 = 0. Solving this quadratic equation, we find that r = 3 is one of the solutions.
Now, using r = 3 in our first equation, S3 = 77 = a + 3a + 9a, we get 77 = 13a, so a = 77/13 = 7. Therefore, the first term is 7 and the common ratio is 3.