Marquan learns a new fact about his favorite animal, the sloth. He shares that fact with two other people. Those two people tell two others, and so on. How many people know the sloth fact after six rounds of information sharing?

Answer 1

To determine how many people know the sloth fact after six rounds of information sharing, we can model the situation as follows:

1. Round 0: Marquan knows the fact. (1 person)

2. Round 1: Marquan shares the fact with 2 other people. Now, there are: \[ 1 + 2 = 3 \text{ people who know the fact.} \]

3. Round 2: Each of those 2 people who learned the fact in Round 1 shares it with 2 others. Therefore, 2 new people learn the fact: \[ 3 + 2 \times 2 = 3 + 4 = 7 \text{ people} \] (Note: In Round 2, the 2 new people are added to the total of 3.)

4. Round 3: The 4 new people (from Round 2) each tell 2 other people: \[ 7 + 4 \times 2 = 7 + 8 = 15 \text{ people} \] (Round 3 adds 8 new learners to the group.)

5. Round 4: The 8 people who learned in Round 3 tell 2 others: \[ 15 + 8 \times 2 = 15 + 16 = 31 \text{ people} \]

6. Round 5: The 16 people from Round 4 each tell 2 other people: \[ 31 + 16 \times 2 = 31 + 32 = 63 \text{ people} \]

7. Round 6: The 32 people from Round 5 share it with 2 other people: \[ 63 + 32 \times 2 = 63 + 64 = 127 \text{ people} \]

Now, after 6 rounds of information sharing, the total number of people who know the sloth fact is:

\[ \boxed{127} \]