After having collected samples along the shore all day long, a marine biologist walks back to her campsite, 2.35 km away. Her dog runs ahead, at twice the speed, reaches the campground and turns around until it meets her again. Then, the dog proceeds to run back to the campground again. This continues until the marine biologist reaches her campground. What distance did the dog run?

(use km as units)

To find the distance the dog ran, we need to calculate the total distance covered during each trip between the campground and the marine biologist.

Let's start by finding the distance covered during the first trip when the dog runs from the campground to meet the biologist. We know that the biologist walks 2.35 km back to the campground, and the dog runs ahead at twice the speed. Therefore, the distance covered by the dog during this first trip is 2.35 km * 2 = 4.7 km.

Now, for subsequent trips, we need to consider that the dog runs to meet the biologist and then has to turn around and run back to the campground again.

During the second trip, the dog covers the same distance as the biologist, so it covers 2.35 km.

During the third trip, the dog covers the distance between the biologist and the campground twice, which is again 2.35 km * 2 = 4.7 km.

This pattern continues, where the dog runs the same distance as the biologist on even-numbered trips (2nd, 4th, 6th, etc.) and twice the distance as the biologist on odd-numbered trips (1st, 3rd, 5th, etc.).

Therefore, to find the total distance the dog ran, we need to sum up the distances covered during each trip. In this case, we have:

4.7 km + 2.35 km + 4.7 km + 2.35 km + ... (repeated pattern)

We can see here that the distances covered by the dog form an arithmetic sequence with a common difference of 2.35 km.

To find the total distance, we can use the formula for the sum of an arithmetic series:

S = (n/2) * (a + l)

Where S is the sum, n is the number of terms, a is the first term, and l is the last term.

In this case, the first term is 4.7 km, the last term is to be determined, and the common difference is 2.35 km.

Since the marine biologist walked a total distance of 2.35 km to get back to the campground, we can determine that this is the last term of the sequence. So l = 2.35 km.

To find the number of terms in the sequence, we divide the total distance walked by the marine biologist by the distance covered during each trip. In this case, it is 2.35 km / 2.35 km = 1.

Now, we can substitute these values into the formula to find the total distance covered by the dog:

S = (n/2) * (a + l)
= (1/2) * (4.7 km + 2.35 km)
= (1/2) * 7.05 km
= 3.525 km

Therefore, the dog ran a total distance of 3.525 km.