Problem 88 - 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: Understand the Pattern

Each generation back, the number of ancestors doubles. For example, 1st generation back (parents) has 2 ancestors, 2nd generation back (grandparents) has 4, and 3rd generation back (great-grandparents) has 8.

Step 2: Identify the Formula

The pattern follows a geometric sequence where each generation is given by the formula: \[ \text{Number of Ancestors} = 2^n \]where \( n \) represents the generation number back from the person.

Step 3: Calculate for 15 Generations

Using the formula, for 15 generations back, substitute \( n = 15 \):\[ \text{Number of Ancestors} = 2^{15} \].

Step 4: Evaluate the Formula

Calculate \( 2^{15} \). This equals to:\[ 2^{15} = 32768 \].

Step 5: Determine Total Ancestors

We need to include all ancestors from generation 1 to 15. So, calculate the sum:\[ 2^1 + 2^2 + 2^3 + \, ... \, + 2^{15} \].

Step 6: Use Geometric Series Formula

The sum of a geometric series is calculated using the formula:\[ S = \frac{a(r^n - 1)}{r - 1} \]Divide by adding all powers of 2 from 1 to 15. Here, \( a = 2 \), \( r = 2 \), and \( n = 15 \).

Step 7: Calculate the Total

Plug in the values into the geometric series formula:\[ S = \frac{2(2^{15} - 1)}{2 - 1} = 2(32768 - 1) = 2 \times 32767 = 65534 \].

Key Concepts

Exponential Growth

Exponential growth is a concept where quantities increase rapidly, at a consistent multiplication rate. In the context of ancestors, each generation doubles the number of individuals from the previous one. This creates a pattern like 2, 4, 8, and so forth.
Each generation you go back adds double the previous number of people. It's a quick way to see rapidly increasing numbers:

• 1st generation (parents): 2 ancestors
• 2nd generation (grandparents): 4 ancestors
• 3rd generation (great-grandparents): 8 ancestors

This pattern illustrates how exponential growth can lead to large numbers very quickly. Understanding how each step builds upon the last helps grasp why numbers grow so fast. Every new level multiplies the ancestral count by 2, leading to exponential escalation.

Generation Pattern

The generation pattern in family trees follows a structured doubling trend. This is because each individual has two parents, producing a clear lineage growth:

• Initially, the generation pattern is established with parents.
• Next, grandparents follow by doubling the parent count.
• Continues with great-grandparents, repeating the doubling process.

The formula to represent this pattern is derived from geometric sequences: \[ \text{Number of Ancestors} = 2^n \]
Here, \( n \) signifies how many generations back you are looking. This specifies that for each shift back in generations, you multiply by 2. Recognizing and applying this pattern is a crucial comprehension in finding the total number of ancestors at any specific generation.

Geometric Series Formula

A geometric sequence forms when each term after the first is found by multiplying the previous one by a fixed, non-zero number. Using a geometric series formula is key in calculating the sum of all generations up to a certain point.

The formula used to sum numbers in our case is: \[ S = \frac{a(r^n - 1)}{r - 1} \]
Breaking it down:

• \( a \) is the initial number which here is 2 (starting number of ancestors)
• \( r \) is the common ratio, in this case, again 2 (doubles every generation)
• \( n \) is the number of terms, which would be 15 for 15 generations

This formula allows you to include the sum of ancestors at each generation, rather than just for one particular generation. Understanding this calculation is pivotal, as it gives the total number of ancestors from generation 1 through 15. By applying the formula, one can quickly find the total at 15 generations to be 65,534 ancestors, demonstrating the power of geometric sequences in analyzing growth patterns over time.