Brick Pile Analysis: Calculating Rows and Bricks

Understanding the structure of a brick pile involves identifying patterns and using mathematical sequences to count the rows and determine the number of bricks in each row. This article will walk you through a step-by-step approach to solve such problems with clear examples and logical reasoning.

Introduction

In a brick pile, the number of bricks in each row decreases uniformly as you move upwards. This creates an arithmetic sequence. The given problem involves determining the number of bricks in the 13th row and the total number of rows in the pile. We will use the properties of arithmetic sequences to solve both parts of the problem.

Part A: Number of Bricks in the 13th Row

Let's start by identifying the pattern in the number of bricks in each row. The number of bricks in each row decreases by 4 as we move up each row, starting with 85 bricks in the bottom row.

The sequence can be described by the formula:

Bricks in (n)-th row 85 - 4(n - 1)

Step-by-Step Solution

Identify the first row and the common difference (difference between consecutive terms): First row: 85 bricks Common difference (d -4) Use the formula for the (n)-th term of an arithmetic sequence: For the 13th row ((n 13)):
/85 - 4(13 - 1)
85 - 4(12)
85 - 48
37 bricks

Thus, there are 37 bricks in the 13th row from the bottom. To find the 13th row from the top, observe that the 13th row from the top is the 10th row from the bottom. We can use the same formula to find this:

Using (n 10):
/85 - 4(10 - 1)
85 - 4(9)
85 - 36
49 bricks

Therefore, there are 49 bricks in the 13th row from the top.

Summary:

37 bricks in the 13th row from the bottom. 49 bricks in the 13th row from the top. 22 total rows in the pile.

Part B: Total Number of Rows in the Pile

Next, we need to determine the total number of rows in the pile. We do this by finding when the number of bricks reaches 1 in the top row. We can use the formula for the (n)-th term of an arithmetic sequence:

85 - 4(n - 1) 1

Step-by-Step Solution

Solve for (n):
85 - 4(n - 1) 1
85 - 4n 4 1
89 - 4n 1
88 4n
n 22

Thus, there are 22 rows in the pile.

Conclusion

Using the properties of arithmetic sequences, we have determined the number of bricks in the 13th row and the total number of rows in the pile. This method can be applied to similar problems involving sequences and series in other mathematical contexts.