Problem 93 - Precalculus with Limits

Question

A brick patio has the approximate shape of a trapezoid (see figure). The patio has \( 18 \) rows of bricks. The first row has \( 14 \) bricks and the \( 18 \)th row has \( 31 \) bricks. How many bricks are in the patio?

Step-by-Step Solution

Answer

There are 405 bricks in the patio.

Step 1: Identify the Values to Use in the Formula

The values are directly given in the problem. The number of terms, \( n \), is the number of rows, which is \( 18 \). The first term, \( a \), corresponds to the number of bricks in the first row, which is \( 14 \). The last term, \( l \), corresponds to the number of bricks in the 18th row, which is \( 31 \).

Step 2: Apply the Formula

Now to find the total number of bricks in the patio, apply the formula for the sum of an arithmetic series: \( \frac{n}{2} * (a + l) \). Substitute \( n = 18 \), \( a = 14 \) and \( l = 31 \) into the formula and solve.

Step 3: Perform the Calculations

Upon substituting, we obtain \( \frac{18}{2} * (14 + 31) = 9 * 45 = 405 \). Therefore, there are 405 bricks in total in the patio.

Key Concepts

Sum of a Series

Understanding the sum of an arithmetic series is essential when trying to find the total number of items arranged with a pattern. An arithmetic series is a sequence of numbers in which each term increases by a constant difference. To find the sum, there's a helpful formula: \[ S_n = \frac{n}{2} \times (a + l) \]where:- \( n \) is the number of terms (in this case, rows of bricks),- \( a \) is the first term (bricks in the first row),- \( l \) is the last term (bricks in the last row).The formula works by calculating the average of the first and last terms, then multiplying by the number of terms. This method takes advantage of the symmetry in arithmetic progressions, ensuring every possible pairing of terms (like the first and last) sums up evenly! With 18 rows, a start of 14 bricks, and an end of 31 bricks, applying this formula gives a total of 405 bricks.

Sequences and Series

When tackling sequences and series, it’s helpful to understand how numbers are arranged and related. A sequence is a list of numbers in a certain order. In arithmetic sequences, every term after the first is obtained by adding a common difference. A series, on the other hand, is the sum of the terms of a sequence. For the patio problem, the bricks in each row form an arithmetic sequence: - First row: 14 bricks - Increment by a common difference (here, the difference is 1, since we don't add it directly but know the first and last values) - Last row: 31 bricks Recognizing this sequence arrangement allows us to apply the sum of a series formula efficiently, as the sequence’s nature simplifies calculations, avoiding the need for listing every single term.

Problem-Solving Strategies

To effectively solve problems involving sequences and series, follow a systematic approach: - **Identify Given Information**: Start by extracting values provided in the problem. For instance, here the number of terms (18 rows), and the values at the start and end of the sequence (14 and 31). - **Choose the Right Formula**: Recognize that a series is involved, and select a relevant formula, such as the sum formula for an arithmetic series. - **Substitute and Solve**: Plug in identified values into your formula. Remember to handle calculations step by step, ensuring each operation there's consistently followed through, like performing arithmetic in stages. These strategies not only simplify the solution process but build a solid understanding of mathematical concepts, allowing you to tackle a variety of problems with confidence.