Problem 72 - Precalculus Mathematics 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?

Step-by-Step Solution

Answer

A person has 65,534 ancestors 15 generations back.

Step 1: Identify the Pattern

The problem specifies that each generation, the number of ancestors doubles. This means that there is a sequence that describes the number of ancestors in each generation, specifically a geometric sequence. Let's represent this: at generation 1 (parents), the number is 2; at generation 2 (grandparents), the number is 4, and so on.

Step 2: Analyze the Sequence

Recognize that the number of ancestors at each generation forms a geometric sequence where the first term \( a = 2 \) (representing 2 parents), and the common ratio \( r = 2 \) (representing that the number of ancestors doubles each generation).

Step 3: Formulate the General Formula

For a geometric sequence, the formula to find the \( n \)-th term is \( a_n = a \, r^{n-1} \). In this case, \( a = 2 \) and \( r = 2 \). Therefore, the formula to find the number of ancestors \( n \) generations back is \( a_n = 2 \times 2^{n-1} \).

Step 4: Calculate the Total Number of Ancestors 15 Generations Back

To find the total number of ancestors after 15 generations, we need to find \( a_1 + a_2 + a_3 + \, ... + \, a_{15} \), since each generation count is distinct and contributes to the sum of all ancestors. This can be represented as \( 2^1 + 2^2 + 2^3 + \, ... \, + 2^{15} \).

Step 5: Use the Sum Formula for a Finite Geometric Series

The sum \( S_n \) of a geometric series \( a + ar + ar^2 + ... + ar^{n-1} \) is given by \( S_n = a \frac{r^n - 1}{r - 1} \). Here, \( a = 2 \), \( r = 2 \), and \( n = 15 \). Thus, the sum of the ancestors is \[ S = 2 \left( \frac{2^{15} - 1}{2 - 1} \right) = 2 \times (2^{15} - 1) \].

Step 6: Compute the Sum

Calculate \( 2^{15} = 32768 \). Therefore, the number of ancestors 15 generations back, including all previous generations, is \( S = 2 \times (32768 - 1) = 2 \times 32767 = 65534 \).

Key Concepts

Finite Geometric Series

A Finite Geometric Series is essentially a sum of terms in a geometric sequence, up to a certain number of terms. Each term in a geometric sequence is derived by multiplying the previous term by a constant. This constant is known as the common ratio. When dealing with a finite sequence, it means there is a definite end to the number of terms you need to add.

In the context of finding out how many ancestors a person has 15 generations back, we need to first recognize that each generation forms part of a geometric sequence. Each generation back represents a new term that needs to be added to the sequence until we have gone back 15 generations.

To calculate the sum of a finite geometric series, you use the formula:

• The initial term, represented by \( a \), which is the number of ancestors at the first generation. In our exercise, \( a = 2 \) because a person has two parents.
• The common ratio, \( r \), which is how much each term is multiplied by to get the next term. Here, \( r = 2 \), as each generation doubles.
• The number of terms in the series, represented by \( n \), which in this case is 15.

Using the sum formula for a finite geometric series:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
you can easily find the total number of ancestors by plugging in the values for \( a \), \( r \), and \( n \). This formula is a powerful tool in simplifying what could be a laborious process of adding individual terms.

Geometric Sequence Formula

At the heart of understanding how generations multiply lies the geometric sequence formula. A geometric sequence is where each term is found by multiplying the previous term by a fixed, constant value, termed as the common ratio. This formula is essential when solving problems related to patterns where growth or change is uniform.

The formula for the nth term of a geometric sequence is given by:
\[ a_n = a \cdot r^{n-1} \]

Let's break this formula down:

• \( a_n \) represents the nth term of the sequence.
• \( a \) is the first term in the sequence. In the generational context, this is the number of parents a person has, which is 2.
• \( r \) is the common ratio. In this ancestry case, it is 2, signifying that each previous generation doubles.
• \( n \) is the number of terms, or generations in this problem.

By applying this formula, one can determine the exact number of ancestors in a particular generation without needing to manually double-count each previous generation. This saves time and reduces complexity. For example, to find the number of ancestors at the 15th generation, substitute \( n \) with 15 in the formula.

Generations and Ancestors

The concept of "Generations and Ancestors" is both fascinating and important in tracing family history or conducting specific genealogical studies. It centers around the notion that each new generation has potentially twice as many ancestors as the one before it. This idea perfectly illustrates a geometric sequence where each term doubles.

A person's direct line of ancestors grows exponentially as you move further back in their lineage. This exponential growth forms a neat geometric sequence, highlighting the power of compound growth, which is often encountered in biology and population studies.

For instance, in our exercise, you start with two parents. Then, each preceding generation sees the number of ancestors double:

• Parents: 2
• Grandparents: 4
• Great-grandparents: 8
• And so on...

In practical terms, this illustrates how vast and complex an individual's heritage can become when traced back over several generations. Understanding how to use geometric sequences aids in quantifying and comprehending this growth. It also shows the importance of each individual within the larger scope of their family's history. The insight into one's lineage can be improved through mathematical understanding, which allows us to see emerging patterns and the true extent of our ancestral roots.