You have two parents, four grandparents, eight great-grandparents, and so forth.

(a)

If all your ancestors were distinct, what would be the total number of your ancestors for the past 42 generations (counting your parents' generation as number one)? (*Hint*: Use the formula for the sum of a geometric sequence.)

To calculate the total number of ancestors over 42 generations (assuming all ancestors are distinct), we can use the sum of a geometric sequence.

### Step-by-Step Breakdown

1. **Each generation's number of ancestors:**
   - You have 2 parents (generation 1),
   - 4 grandparents (generation 2),
   - 8 great-grandparents (generation 3), and so on.
   - This forms a geometric sequence where:
     - The first term \( a = 2 \) (your parents),
     - The common ratio \( r = 2 \) (each generation has twice as many ancestors as the previous),
     - The number of generations \( n = 42 \).

2. **Geometric sequence formula:**
   The sum \( S_n \) of the first \( n \) terms of a geometric sequence is given by:
   \[
   S_n = a \frac{r^n - 1}{r - 1}
   \]
   Substituting the values \( a = 2 \), \( r = 2 \), and \( n = 42 \):
   \[
   S_{42} = 2 \cdot \frac{2^{42} - 1}{2 - 1}
   \]
   \[
   S_{42} = 2 \cdot (2^{42} - 1)
   \]
   
3. **Calculate \( 2^{42} \):**
   \[
   2^{42} = 4398046511104
   \]
   So,
   \[
   S_{42} = 2 \cdot (4398046511104 - 1) = 2 \cdot 4398046511103 = 8796093022206
   \]

### Final Answer:
The total number of ancestors for the past 42 generations is \( 8,796,093,022,206 \).

Would you like more details or have any questions?

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Here are five related questions for further exploration:

1. What would be the total number of ancestors for 50 generations?
2. How does the number of ancestors change if we assume overlapping lineage (some ancestors are repeated)?
3. Can you calculate the total number of ancestors for 30 generations?
4. What happens to this geometric model in real-world population genetics?
5. How many ancestors would you have in the 10th generation?

**Tip:** Geometric sequences grow exponentially, meaning small changes in the number of generations have a massive impact on the total number of ancestors!

You have two parents, four grandparents, eight great-grandparents, and so forth. If all your ancestors were distinct, what would be the total number of your ancestors for the past 42 generations (counting your parents' generation as number one)?

Learn how to calculate the total number of distinct ancestors over 42 generations using the sum of a geometric sequence. This problem involves exponential growth and geometric sequences, providing a step-by-step solution. Suitable for high school students in Grades 9-12.

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