Problem 64 - Precalculus: Mathematical for Calculus

Question

A person has two parents, four grandparents, eight great-grandparents, and so on. How many ancestors does a person have 15 generations back? (GRAPH CANT COPY)

Step-by-Step Solution

Answer

A person has 65534 ancestors up to 15 generations back.

Step 1: Understanding the Problem

We are asked to find the total number of ancestors a person has up to the 15th generation. Each generation back doubles the number of ancestors compared to the previous generation.

Step 2: Identifying the Pattern

Notice that each generation increases the number of ancestors by a factor of 2. For example, 1st generation has 2 ancestors, 2nd generation has 4 ancestors, 3rd generation has 8, and so on. This forms a geometric sequence.

Step 3: Identifying the Formula

The number of ancestors in the nth generation can be expressed as: \[ 2^n \] where \( n \) is the generation number. To find the total number of ancestors up to and including the 15th generation, we need to sum these terms.

Step 4: Calculating the Geometric Series

The sum of ancestors from 1st to 15th generation is calculated as the sum of the geometric series: \[ S = 2^1 + 2^2 + 2^3 + \, ... \, + 2^{15} \]. This series can be simplified using the formula for the sum of a geometric series: \[ S = a \left( \frac{r^n - 1}{r - 1} \right) \], with \( a = 2 \), \( r = 2 \), and \( n = 15 \).

Step 5: Applying the Geometric Series Formula

Substitute into the formula: \[ S = 2 \left( \frac{2^{15} - 1}{2 - 1} \right) = 2^{16} - 2 \]. Calculate: \( 2^{16} = 65536 \), so \( S = 65536 - 2 = 65534 \).

Step 6: Result Interpretation

The total number of ancestors up to and including the 15th generation is 65534.

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." For example, in the context of ancestral generations, each generation back doubles the number of ancestors, which gives us a common ratio of 2. Therefore, starting with the first generation where you have 2 parents, the second generation has 4 grandparents (2 times 2), the third generation has 8 great-grandparents (4 times 2), and so on.
This predictable pattern is a hallmark of geometric sequences:

• First generation: 2 ancestors (2¹)
• Second generation: 4 ancestors (2²)
• Third generation: 8 ancestors (2³)
• n-th generation: 2ⁿ ancestors

This mathematical structure allows us to apply formulas easily to solve broader problems, like finding the total number of ancestors across multiple generations.

Exponential Growth

Exponential growth is a process that increases quantity over time. It happens when the growth rate of a value is proportional to the current value, causing the quantity to grow at an increasingly rapid rate. This is exactly what occurs in a geometric sequence when each term is multiplied by a constant factor, like in our ancestor example where each generation doubles.
Consider the process of calculating generational ancestors:

• The first generation consists of 2 ancestors.
• By the time you reach the 15th generation, the ancestors grow exponentially, calculated by using the formula for exponential growth: \[ X = a \cdot r^n \]

Where:- \(X\) is the final quantity (number of ancestors)- \(a\) is the initial quantity (2 in this case)- \(r\) is the growth rate (2 for doubling each generation)- \(n\) is the number of times the growth occurs (in our example, 15 generations)This kind of growth is not only mind-boggling in theory but also highlights the amazing nature of exponential patterns we observe in various contexts.

Generational Calculations

Generational calculations in the context of ancestry involve determining the total number of ancestors a person has across several generations. This involves the summation of a geometric series, where each generation contributes more ancestors than the previous one due to exponential growth. In our exercise, the goal is to find the total number of ancestors from the first to the 15th generation.
To carry out this calculation, we use the sum formula of the geometric series:\[S = a \left( \frac{r^n - 1}{r - 1} \right)\]Here, \(a\) is the first term of the sequence, \(r\) is the common ratio, and \(n\) is the number of terms.

• \(a = 2\) because the first generation has 2 ancestors.
• \(r = 2\) because each generation doubles.
• \(n = 15\) representing 15 generations.

So, when applying these values into the formula, the result for the total number of ancestors is calculated as \(2 (2^{15} - 1)/(2 - 1)\), which simplifies to \(2^{16} - 2\). This gives 65534 ancestors. Understanding this sum helps us appreciate how rapidly the number of ancestors increases as we consider more generations.