Problem 64 - 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?

Step-by-Step Solution

Answer

A person has 65,534 ancestors 15 generations back.

Step 1: Understanding Generational Count

Each person has 2 parents. Continuing this pattern, each generation doubles the count from the previous generation. If we start from generation 0 (the person themselves) and move to generation 1, they have 2 ancestors, for generation 2 they have 4, and it continues doubling.

Step 2: Establishing the Formula

We can establish that the count of ancestors in a specific generation is given by the formula \(2^n\), where \(n\) is the generation number. Thus, the first generation (parents) is \(2^1 = 2\), the second generation (grandparents) is \(2^2 = 4\), and so forth.

Step 3: Calculating Total Ancestors for 15 Generations

To find the number of ancestors across all generations up to 15, we use the sum formula for the ancestors, starting from generation 1 to generation 15: \(2^1 + 2^2 + 2^3 + \, ... \, + 2^{15}\). This is a geometric series.

Step 4: Applying Geometric Series Formula

The sum of a geometric series can be calculated using the formula: \( S_n = a \times \frac{r^n - 1}{r - 1} \), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. For this series, \(a = 2^1\), \(r = 2\), and \(n = 15\).

Step 5: Calculate the Sum

Using the formula, we substitute \(a = 2\), \(r = 2\), and \(n = 15\): \[ S_{15} = 2 \times \frac{2^{15} - 1}{2 - 1} = 2 \times (2^{15} - 1) \] Calculate: \[ 2^{15} = 32768 \] \[ S_{15} = 2 \times (32768 - 1) = 2 \times 32767 = 65534 \]

Key Concepts

Generation Doubling

When we talk about generation doubling, it’s a concept where each subsequent generation multiplies by two compared to the previous generation. This is because every individual has two parents, which forms the basis of a binary family tree. Think of it as each generation increasing like branches from a main trunk. So, for generation 0 (the person themselves), there are zero additional individuals in the generation chart.
For every generation, you move back; for example:

• Generation 1 (the parents) introduces 2 people.
• Generation 2 (the grandparents) adds 4 people.
• Generation 3 (the great-grandparents) adds 8 people.

This pattern continues exponentially as you go further back, each generation holds twice the number of ancestors as the previous one. Understanding this pattern helps in calculating large family trees easily.

Exponential Growth

Exponential growth refers to quantities increasing at a rate that is proportional to its current value, leading to quicker and quicker growth. In the genealogy context, it's illustrated through the increasing number of ancestors as we go back through generations.
Each generation shows an increasing number of ancestors by a factor of two, doubling the family size each time. For example, having just a couple of ancestors in generation 1 quickly transforms into thousands by generation 15.
This type of growth is seen in various real-world applications, such as:

• Population growth
• Compound interest in finance
• Biological processes, such as bacteria multiplying

All these examples rely on the core principle of each iteration based on the current state rather than a fixed increment.

Geometric Series Formula

The geometric series formula is a handy mathematical tool to sum a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In our case with generations, this is seen in counting the total number of ancestors up to a certain point.
The formula is expressed as:\[S_n = a \times \frac{r^n - 1}{r - 1}\]where:

• \(S_n\) is the sum of the series
• \(a\) is the first term of the series
• \(r\) is the common ratio between terms
• \(n\) is the number of terms

By applying this to the family tree exercise, we calculate how many ancestors one would have spanning from generation 1 to generation 15, using 2 as the common ratio since each generation doubles. This shows how mathematical formulas like this are incredibly effective in solving real-life problems efficiently.