Problem 72 - Algebra and Trigonometry

Question

Family Tree A person has two parents, four grandparents, eight great- grandparents, and so on. How many ancestors does a person have 15 generations back? Graph cannot copy

Step-by-Step Solution

Answer

A person has 65,534 ancestors 15 generations back.

Step 1: Understand the Problem

In this problem, we are asked to find out how many ancestors a person has going back 15 generations. Each generation back, the number of ancestors doubles.

Step 2: Identify the Pattern

Each generation has double the number of ancestors as the previous one. So, the number of ancestors in generation \( n \) is given by \( 2^{n} \).

Step 3: Calculate for 15 Generations

To find the number of ancestors 15 generations back, we calculate \( 2^{15} \).

Step 4: Compute 2 to the Power of 15

Calculate \( 2^{15} \) which equals 32,768. This means there are 32,768 ancestors in the 15th generation.

Step 5: Sum the Ancestors

To find the total number of ancestors from generation 1 to 15, sum up all individuals from each generation: \(2^1 + 2^2 + 2^3 + \, ... \, + 2^{15}\). This is a geometric series.

Step 6: Calculate the Sum of the Geometric Series

The sum of a geometric series is given by \( S_n = a(1-r^n)/(1-r) \), where \( a = 2 \), \( r = 2 \), and \( n = 15 \). Calculate \[ S_{15} = 2 \frac{1-2^{15}}{1-2} = 2(32,768 - 1) = 65,534. \]

Step 7: Conclusion

A person has a total of 65,534 ancestors going 15 generations back.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a specific factor, known as the common ratio. For example, if the first term is 2 and the common ratio is also 2, the sequence will be 2, 4, 8, 16, and so on. The beauty of a geometric series is its predictability, which is handy for calculations, like in ancestry.

• Each term is derived by multiplying the previous term by the common ratio.
• The formula for the sum of a geometric series is \( S_n = a\frac{1-r^n}{1-r} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

In the context of ancestry, the "common ratio" of 2 signifies that each generation effectively doubles the number of ancestors.

Power of Two

The power of two is a fundamental concept in mathematics, especially when dealing with problems related to doubling and binary systems. The expression \( 2^n \) means 2 multiplied by itself \( n \) times. Understanding powers of two is crucial when examining phenomena like exponential growth in ancestry.

• \( 2^1 = 2 \), \( 2^2 = 4 \), \( 2^3 = 8 \), and so on.
• It represents a sequence that grows rapidly, which is characteristic of exponential growth.

In an ancestry problem, \( 2^n \) symbolizes the number of individuals in the \( n^{th} \) generation, leading to large numbers over multiple generations: \( 2^{15} = 32,768 \). This progression highlights the vast number of ancestors possible within just a few generations.

Generations in Ancestry

Generations in ancestry follow a predictable pattern where the number of ancestors doubles with each further generation. This stems from the fact that each person has two biological parents, leading to a binary branching pattern.

• Starting with yourself, you have 2 parents, 4 grandparents, 8 great-grandparents, and so forth.
• This pattern aligns perfectly with the formula \( 2^n \) for thee number of ancestors at \( n^{th} \) generation.
• Understanding this helps us visualize and calculate distant genealogical connections effectively.

For example, 15 generations back, you would have \( 2^{15} = 32,768 \) individuals in your family tree if we just follow the direct parental line without considering any overlaps or repeats.

Mathematical Pattern Recognition

Mathematical pattern recognition is crucial in understanding ancestry problems and predicting outcomes. Recognizing patterns simplifies complex calculations and enables us to make general predictions based on the rules of mathematics.

• By examining initial generations, you see a repetitive sequence of doubling.
• This doubling is recognized as a mathematical pattern represented by powers of two.
• The sum of these patterns, when extended, forms a geometric series, allowing quick calculation of sums over multiple generations.

For instance, through pattern recognition, one can swiftly calculate total ancestors over a span of generations by recognizing the geometric series as summarized in the formula \( S_n = a\frac{1-r^n}{1-r} \), which succinctly handles vast quantities of numbers.