How many grandparents are there in 10 generations?
Understand the Problem
The question is asking for the total number of grandparents over a span of 10 generations, highlighting the exponential growth of family lineage as each individual can have two parents.
Answer
$2046$
Answer for screen readers
The total number of grandparents over 10 generations is $2046$.
Steps to Solve
- Understand the Family Tree Growth
Each person has two parents. Thus, for each generation back, the number of individuals doubles.
- 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 $2^2 = 4$.
- Apply the Formula for 10 Generations
To find the total number at the 10th generation: $$ 2^{10} = 1024 $$
- 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 $10$).
- Plug In the Values to Find the Total Sum
Using the formula: $$ S_{10} = 2 \frac{(2^{10} - 1)}{(2 - 1)} = 2 (2^{10} - 1) = 2 (1024 - 1) = 2 \times 1023 = 2046 $$
The total number of grandparents over 10 generations is $2046$.
More Information
This calculation shows how rapidly family lineage expands. Each generation greatly increases the potential number of ancestors due to the doubling effect from having two parents.
Tips
- Forgetting to sum all generations: Some might only focus on the last generation and forget to consider all previous generations.
- Incorrectly applying the geometric sum formula: Ensure you correctly identify the first term and the common ratio.