Problem 58 - Contemporary Precalculus

Question

A monument is constructed by first laying a row of 60 bricks at ground level. A second row, with two fewer bricks, is centered on that; a third row, with two fewer bricks, is centered on the second; and so on. The top row contains 10 bricks. How many bricks are there in the monument?

Step-by-Step Solution

Answer

Answer: There are 1820 bricks used to construct the monument.

Step 1: Identify the arithmetic series

In this situation, the arithmetic series is the number of bricks in each row, starting with the first row (60 bricks) and ending with the last row (10 bricks). The difference between each consecutive term (number of bricks in each row) is -2. Series: 60, 58, 56, ..., 12, 10 Common difference: -2

Step 2: Find the number of terms (rows) in the series

Let n be the number of terms in the series. We will use the arithmetic series formula: last term = first term + (n - 1) * common difference In our case, the first term is 60, the last term is 10, and the common difference is -2. We will solve for n. 10 = 60 + (n - 1) * (-2)

Step 3: Solve for n

To find the number of terms in the series, solve the equation from Step 2. 10 = 60 - 2n + 2 2n = 60 + 2 - 10 2n = 52 n = 26 So, there are 26 terms (rows) in the series.

Step 4: Find the total number of bricks in the monument

Now we will use the sum formula for arithmetic series to find the total number of bricks in the monument. The formula is: Sum = (n * (first term + last term)) / 2 In our case, n is 26, the first term is 60, and the last term is 10. Sum = (26 * (60 + 10)) / 2

Step 5: Calculate the sum

Now, we will calculate the sum: Sum = (26 * 70) / 2 Sum = 1820 So, there are 1820 bricks in the monument.

Key Concepts

Arithmetic Series Formula

The arithmetic series formula is a fundamental concept in precalculus, crucial for solving problems involving the summation of terms in a sequence where each term is generated by adding a constant value to the previous term. This constant value is known as the common difference. The series formula is expressed as \[\text{Sum} = \frac{n}{2} \times (\text{first term} + \text{last term})\], where 'n' represents the number of terms in the series.

Let's apply this concept to our monument of bricks. If you look at each row as a term in an arithmetic series, you can see a pattern: each row has two fewer bricks than the row below it, which means our common difference is -2. Using the formula for arithmetic series, we can calculate the total number of bricks by adding the first term (60 bricks) to the last term (10 bricks), multiplying by the number of terms, and then dividing by 2.

Sum of Arithmetic Series

Understanding the sum of an arithmetic series is crucial when we want to find the total of all terms added together. The sum is given by the formula previously mentioned. However, in practice, it involves identifying the series' characteristics: the first term, the last term, and the number of terms. Calculating the sum helps in situations like our brick monument where we want to know the total count of bricks without having to add each term individually.

In our exercise, after determining that there are 26 rows using the given common difference and the first and last terms, we apply the sum formula. By plugging these values into the sum formula, we obtain \[\text{Sum} = (26 \times (60 + 10)) / 2\], which simplifies to 1820 bricks in total. It's a streamlined way to calculate the sum without individually counting each brick in every row.

Arithmetic Sequences

Arithmetic sequences are a string of numbers where each term after the first is created by adding a constant value, known as the common difference, to the previous term. The sequence can be thought of as a list of elements with well-defined ordering. The common difference could be positive or negative, leading to an increasing or decreasing sequence respectively.

For the brick monument, the sequence starts at 60 (first row) and decreases by 2 bricks per row, forming an arithmetic sequence: 60, 58, 56, ..., 12, 10. This sequence embodies how each layer of the monument builds upon the previous one by a specific rule, dictated by the common difference. Recognizing this pattern allows for the entire structure to be visualized and analyzed mathematically, which in turn simplifies finding the total number of bricks that make up the monument.