How many grandparents are there in 10 generations?

Steps to Solve

1. Understand the Family Tree Growth

Each person has two parents. Thus, for each generation back, the number of individuals doubles.

2. Calculate the Total Number in Each Generation

The formula to determine the number of grandparents at any generation $n$ is $2^n$. For example:

• In the 1st generation (parents), there are $2^1 = 2$.
• In the 2nd generation (grandparents), there are $2^2 = 4$.

3. Apply the Formula for 10 Generations

To find the total number at the 10th generation: $$ 2^{10} = 1024 $$

4. Calculate the Total Number of Grandparents in 10 Generations

The total number of grandparents over all generations is the sum of all individuals from the 1st to the 10th generation: $$ 2^1 + 2^2 + 2^3 + ... + 2^{10} $$

This can be simplified using the formula for the sum of a geometric series: $$ S_n = a \frac{(r^n - 1)}{(r - 1)} $$ Where $a$ is the first term (which is $2$), $r$ is the common ratio (which is also $2$), and $n$ is the number of terms (which is $10$).

5. Plug In the Values to Find the Total Sum

Using the formula: $$ S_{10} = 2 \frac{(2^{10} - 1)}{(2 - 1)} = 2 (2^{10} - 1) = 2 (1024 - 1) = 2 \times 1023 = 2046 $$