A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.

Step 1: Identify the pattern

The number of ancestors in each preceding generation forms a geometric progression. The first generation (parents) has 2 ancestors, the second (grandparents) has 4, and the third (great grandparents) has 8. This means each term is obtained by multiplying the previous term by 2.

Step 2: Determine the common ratio and first term

From the identified pattern, the first term $a$ (number of parents) is 2. The common ratio $r$ is found by dividing any term by its preceding term, for example, $4/2 = 2$ or $8/4 = 2$.

Step 3: Identify the number of terms

We need to find the total number of ancestors for the ten generations preceding the person's own. Therefore, the number of terms in this series is 10.

Step 4: Apply the sum of a geometric series formula

To find the total number of ancestors, we need to sum the number of ancestors in each of the 10 generations. The formula for the sum of the first $n$ terms of a geometric series is $S_n = \frac{a(r^n - 1)}{r - 1}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Step 5: Substitute values and calculate the sum

Substitute the values $a=2$, $r=2$, and $n=10$ into the sum formula. First, calculate $2^{10}$ which is 1024. Then, subtract 1, multiply by 2, and divide by $2-1=1$ to get the final sum.

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