Question

A person has two parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.

• A. 2042
• B. 2044
• C. 2046
• D. 2048

Hint:

To calculate the total number of ancestors in \( n \) generations, use the sum of the geometric series: \( 2^1 + 2^2 + \dots + 2^n \).

Answer 1

The Correct Option is C

Step 1: Understanding the number of ancestors.
Each generation doubles the number of ancestors from the previous generation. The total number of ancestors in \( n \) generations is given by: \[ \text{Total ancestors} = 2^1 + 2^2 + 2^3 + \dots + 2^{10} \]
Step 2: Applying the formula.
This is a geometric series with the first term \( 2^1 \) and the common ratio of 2. The sum of the first \( n \) terms of a geometric series is: \[ S_n = a \frac{r^n - 1}{r - 1} \] where \( a = 2^1 \), \( r = 2 \), and \( n = 10 \). So, the sum is: \[ S_{10} = 2 \frac{2^{10} - 1}{2 - 1} = 2 \times (1024 - 1) = 2 \times 1023 = 2046 \]
Step 3: Conclusion.
Thus, the number of ancestors during the ten generations is 2046. Final Answer:} 2046.