Problem 76 - Beginning and Intermediate Algebra

Question

A landscaper plans to put a pyramid design in a brick patio so that the bottom row of the pyramid contains 9 bricks and every row above it contains two fewer bricks. How many bricks does she need to make the design?

Step-by-Step Solution

Answer

The landscaper needs a total of 25 bricks to create the pyramid design in the brick patio, with 5 rows following the pattern of 9, 7, 5, 3, and 1 bricks per row.

Step 1: Identify the pattern

As stated in the exercise, we know that the pyramid has a pattern where each row has 2 fewer bricks than the row below it. So, starting from the bottom row where there are 9 bricks, we can list down the number of bricks in each subsequent row: 9, 7, 5, 3, 1.

Step 2: Calculate the number of rows

Based on the pattern, we've just listed 5 rows in the pyramid (9, 7, 5, 3, 1). Therefore, there are 5 rows in the pyramid.

Step 3: Determine the total number of bricks

Now, to find out the total number of bricks in the pyramid, all we need to do is add the number of bricks in each row: 9 + 7 + 5 + 3 + 1 = 25 bricks So, the landscaper needs 25 bricks to make the pyramid design in the brick patio.

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two successive terms is constant. In simpler terms, you add or subtract the same number to each term to get to the next term. This constant difference is known as the common difference. For example, if you take the sequence 2, 4, 6, 8, the common difference here is 2, because each number is 2 more than the previous one.
In the pyramid brick design, the landscaper uses an arithmetic sequence where each row has 2 fewer bricks than the previous row. So, starting with 9 bricks in the bottom row, the sequence would be expressed as 9, 7, 5, 3, and 1. The common difference here is -2, indicating a reduction in the number of bricks per row.

Calculating the Sum of a Sequence

When it comes to finding out how many bricks are needed in total, we calculate the sum of the sequence. The sum of an arithmetic sequence can be obtained through a straightforward method of simply adding all the terms.
For small sequences, manual addition is feasible as shown in the exercise with the sequence 9, 7, 5, 3, and 1. Adding these numbers gives us the total number of bricks, which is 25. However, for longer sequences, a formula could be more useful. This formula is \[ S_n = \frac{n}{2} \times (a_1 + a_n) \] where:

• \( S_n \) is the sum of the sequence,
• \( n \) is the number of terms,
• \( a_1 \) is the first term,
• \( a_n \) is the last term.

In our case, though, just adding is simple and effective.

Problem Solving with Sequences

Problem solving using sequences involves recognizing patterns and applying them to find solutions. The task of determining the number of bricks needed for the landscaper's pyramid design is a good example. Breaking down the problem into steps is vital:
First, you recognize that there is a pattern of decreasing bricks per row, which is a characteristic of an arithmetic sequence. Then, follow the subsequent steps of listing the bricks in each row and summing them up.

• Identify the pattern.
• Count the rows and list the terms in the sequence.
• Add the terms to find the total.

By breaking down complex problems into smaller, more manageable parts, solving them becomes straightforward and more approachable.

Recognizing Patterns in Sequences

Patterns are fundamental in sequences and are often the key to unlocking their structure. Identifying the right pattern helps in predicting future terms or summing up the sequence effectively.
In the brick pyramid example, the pattern is clearly a simple arithmetic sequence, where each level of the pyramid hosts two fewer bricks than the level below it. Recognizing this pattern lets you continue the sequence logically.
You can practice spotting patterns by looking for:

• Repetitions or regular changes like increase or decrease by a fixed number.
• Using visual or systematic approaches, like drawing out the sequence or writing it in list form.

Understanding these patterns makes more complex sequences manageable and is an essential mathematical skill.