Ms. Tina has 2 parents, 4 grand parents, 8 great grand parents and so on. Find the number of her ancestors during 10 (ten) generations preceeding her own.

Question

Prepp Team · Accepted Answer

Understanding the Ancestor Calculation Problem

The question asks us to find the total number of ancestors Ms. Tina had in the 10 generations immediately preceding her own generation. We are given the number of ancestors for the first few generations:

Generation 1 (Parents): 2 ancestors
Generation 2 (Grandparents): 4 ancestors
Generation 3 (Great Grandparents): 8 ancestors

This pattern shows that the number of ancestors doubles with each preceding generation. This is a classic example of a geometric progression.

Identifying the Geometric Progression

A geometric progression (GP) is a sequence 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. In this case, the sequence of ancestors per generation is 2, 4, 8, ...

The first term ($a$) is the number of parents, which is 2.
The common ratio ($r$) is the factor by which the number of ancestors increases each generation. Since 4/2 = 2 and 8/4 = 2, the common ratio is 2.
We need to find the total number of ancestors over 10 generations. So, the number of terms ($n$) in the series is 10.

Calculating the Total Number of Ancestors

We need to find the sum of the first 10 terms of this geometric progression. The formula for the sum ($S_n$) of the first $n$ terms of a geometric progression is:

$$S_n = a \frac{(r^n - 1)}{(r - 1)}$$

Where:

$a = 2$ (first term)
$r = 2$ (common ratio)
$n = 10$ (number of generations)

Now, let's substitute these values into the formula:

$$S_{10} = 2 \frac{(2^{10} - 1)}{(2 - 1)}$$

First, calculate $2^{10}$:

$$2^{10} = 1024$$

Next, substitute this value back into the formula:

$$S_{10} = 2 \frac{(1024 - 1)}{(2 - 1)}$$

Simplify the expression:

$$S_{10} = 2 \frac{(1023)}{1}$$

$$S_{10} = 2 \times 1023$$

$$S_{10} = 2046$$

So, the total number of ancestors Ms. Tina had during the 10 generations preceding her own is 2046.

Summary of Calculation Steps

Identify the sequence of ancestors per generation: 2, 4, 8, ...
Recognize this sequence as a geometric progression.
Determine the first term ($a=2$) and the common ratio ($r=2$).
Identify the number of terms ($n=10$) as the number of generations.
Use the sum of a geometric series formula: $$S_n = a \frac{(r^n - 1)}{(r - 1)}$$
Substitute the values: $$S_{10} = 2 \frac{(2^{10} - 1)}{(2 - 1)}$$
Calculate $2^{10} = 1024$.
Complete the calculation: $$S_{10} = 2 \frac{(1024 - 1)}{1} = 2 \times 1023 = 2046$$

The total number of ancestors in 10 generations is 2046.

Revision Table: Ancestor Calculation Concepts

Concept	Description	Application in Problem
Ancestor Generations	Tracing lineage backwards; number typically doubles each generation.	The core problem involves counting ancestors across generations.
Geometric Progression	A sequence where each term is found by multiplying the previous term by a constant ratio.	The number of ancestors (2, 4, 8, ...) forms a GP.
First Term ($a$)	The initial value of the sequence.	Number of parents (2).
Common Ratio ($r$)	The constant multiplier between consecutive terms.	The factor by which ancestors multiply each generation (2).
Number of Terms ($n$)	The total count of elements in the series we are summing.	The number of preceding generations (10).
Sum of GP ($S_n$)	The total value obtained by adding the first $n$ terms of the GP.	The total number of ancestors across all 10 generations.

Additional Information: Geometric Series and Growth

This problem demonstrates exponential growth, where a quantity increases by a constant factor over equal intervals. The number of ancestors grows exponentially as you go back in time.

The formula for the sum of a geometric series is very useful in various applications, including:

Calculating compound interest.
Modeling population growth or decay (if the rate is constant).
Analyzing the value of annuities.
Solving problems involving sequences where quantities double or halve repeatedly.

It's important to note that while this mathematical model is useful for illustrating the concept of exponential growth in ancestry, real-world ancestry can be slightly more complex due to factors like intermarriage within families over many generations, which would slightly reduce the number of unique ancestors.