Problem 60 - Contemporary Precalculus

Question

Starting with your parents, how many ancestors do you have for the preceding ten generations?

Step-by-Step Solution

Answer

Answer: 1,023

Step 1: Generation 1: Parents

In the first generation, there are 2 ancestors: one's mother and father.

Step 2: Generation 2: Grandparents

In the second generation, there are 4 ancestors: each parent has a mother and father, so 2 × 2 = 4.

Step 3: Generation 3: Great-Grandparents

In the third generation, there are 8 ancestors: Each of the 4 grandparents has a mother and father, so 4 × 2 = 8.

Step 4: Iterative Calculation for Generations 4-10

We can continue this pattern for each of the remaining generations. For each generation, the number of ancestors doubles.

Step 5: Using the Formula for summing powers of 2

To calculate the total number of ancestors over the ten generations, we can sum the powers of 2, from 2^1 (parents) to 2^10: \( \sum_{n=1}^{10} 2^n = 2^1 + 2^2 + 2^3 + \cdots + 2^{10}\)

Step 6: Applying the geometric series formula

The sum of the geometric series formula is: \(S_{n} = \frac{1-r^n}{1-r}\) Where \(S_n\) is the sum of the first n terms, r is the common ratio, and n is the number of terms. In our problem, r = 2 and n = 10. \(S_{10} = \frac{1-2^{10}}{1-2}\)

Step 7: Calculate the total number of ancestors

Evaluate the sum: \(S_{10} = \frac{1-1024}{-1} = 1023\) So, there are a total of 1,023 ancestors in the preceding ten generations.

Key Concepts

Geometric Series

Imagine a pattern where each term is a multiple of the previous one – that's the essence of a geometric series. It is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Think of it like this: if you have a single apple and double the amount each day, in ten days, you won’t just have ten apples, but over a thousand! That’s the power of geometric progression.

Now, consider the ancestral problem. The number of ancestors each generation has can be seen as a geometric series where the common ratio, denoted by 'r', is 2 because each generation has twice as many as the previous one. Using the geometric series formula, we can quickly calculate the total number of ancestors across multiple generations. This saves us from having to add each term individually, which is crucial when the number of generations is large.

Let's make this practical: the formula for the sum of a geometric series is given by
\(S_{n} = \frac{1-r^n}{1-r}\).

Exponential Growth

Exponential growth is astounding in its speed – it's like a snowball rolling down a hill, gathering more and more snow at an ever-increasing rate. In mathematical terms, it describes a process that increases in quantity by a consistent relative rate over equidistant periods. In the case of ancestral generations, think of it as a family tree that expands rapidly with each new generation.

Each successive generation doubles, which is a classic example of exponential growth, because the number of ancestors at each step is a multiple of the previous count (in this case, multiplied by 2). This pattern of growth shows how quickly quantities can grow over time – and it's why, after 10 generations, the number doesn't just grow by tenfold but by much, much more.

In real life, exponential growth can apply to populations, finances, or any scenario where quantities compound over time, leading to a rapid increase.

Sum of Powers of 2

The sum of powers of 2 is a fascinating sequence where each term exponentially increases, multiplying by 2 each time. It starts with 2^1 (or simply 2), then 2^2 (which is 4), and so on. To visualize it, picture a chess board with one grain of rice on the first square, two on the second, four on the third, and keep doubling for each of the 64 squares – by the end, you'll have a mountain of rice!

In the context of ancestral generations, the number of ancestors in each generation follows this exponential pattern, represented with powers of 2. Thus, the total count of ancestors over a number of generations can be seen as a sum of powers of 2, starting from 2^1 for the first generation (your parents), and increasing until 2^10 for the tenth generation. This is useful in mathematics as it shows the immense increase that can occur even over a relatively small number of terms, which can be applied to theoretical and practical scenarios alike.