3. carla traced her family tree back 4 generations. carlas parents are the first generation back and her first set of ancestors. carlas four grandparents are the second generatino back and her second set of ancestors.
a) how many ancestors are in carlas family tree? (4 sets)
b) determine how many generations back she would have to trace to find more than 1 million ancestors

### a) **How many ancestors are in Carla's family tree (4 sets)?**

In each generation, the number of ancestors doubles:

1. **First generation**: Carla has 2 parents.
2. **Second generation**: She has 4 grandparents.
3. **Third generation**: She has 8 great-grandparents.
4. **Fourth generation**: She has 16 great-great-grandparents.

To find the total number of ancestors across the four generations, we sum the number of ancestors in each generation:

\[
2 + 4 + 8 + 16 = 30
\]

So, Carla has **30 ancestors** in her family tree across 4 generations.

---

### b) **How many generations back would Carla have to trace to find more than 1 million ancestors?**

The number of ancestors in each generation follows a pattern of doubling every generation. The formula for the number of ancestors in the \( n \)-th generation is \( 2^n \). To find when the total number of ancestors exceeds 1 million, we need to sum the number of ancestors from each generation and find when this sum exceeds 1,000,000.

The sum of ancestors up to the \( n \)-th generation is given by the sum of powers of 2:

\[
2^1 + 2^2 + 2^3 + \dots + 2^n = 2^{n+1} - 2
\]

We need to solve for \( n \) when this total exceeds 1,000,000:

\[
2^{n+1} - 2 > 1,000,000
\]

\[
2^{n+1} > 1,000,002
\]

Taking the logarithm base 2 of both sides:

\[
n+1 > \log_2(1,000,002)
\]

We can now compute the value of \( n \).

Let me calculate that for you.From the calculation, we have \( n+1 > 19.93 \), so \( n+1 \) rounds up to 20. This means \( n = 19 \).

Thus, Carla would have to trace back **19 generations** to find more than 1 million ancestors.

---

Would you like more details or have any further questions?

Here are some related questions you might find interesting:

1. How is the doubling pattern in ancestry related to exponential growth?
2. What is the significance of the formula \( 2^n \) in ancestry problems?
3. How many ancestors would Carla have after 10 generations?
4. How would environmental or historical factors affect the accuracy of this ancestral growth?
5. How does this model change when accounting for potential overlap in family trees (due to distant relations)?

**Tip**: When dealing with powers of 2, small increases in the exponent can lead to large increases in the result, demonstrating exponential growth.

Carla traced her family tree back 4 generations. Carla's parents are the first generation back and her first set of ancestors. Carla's four grandparents are the second generation back and her second set of ancestors. a) How many ancestors are in Carla's family tree (4 sets)? b) Determine how many generations back she would have to trace to find more than 1 million ancestors.

Explore how to calculate the number of ancestors in a family tree using exponential growth. This problem demonstrates the doubling pattern in ancestry over generations and uses logarithms to find how many generations back more than 1 million ancestors exist. Ideal for students in Grades 9-12.

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