Problem 59 - Discrete Mathematics with Applications

Question

Let \(a_{n}\) denote the number of rectangles that can be formed on a \(1 \times n\) rectangular board. Find the recurrence relation satisfied by \(a_{n}\) (Hint: Look for a pattern. Every square is also a rectangle.)

Step-by-Step Solution

Answer

The recurrence relation satisfied by \(a_n\) is \(a_n = a_{n-1} + n\), with the base case \(a_1 = 1\).

Step 1: Observations in smaller cases

Let's start by looking at the case of a \(1 \times 1\) board. There's one possible rectangle (the square itself). Now, for a \(1 \times 2\) board, we have: - 2 rectangles of dimensions \(1 \times 1\) (both squares) - 1 rectangle of dimensions \(1 \times 2\) So, there are a total of 3 rectangles on a \(1 \times 2\) board. Let's check the case of a \(1 \times 3\) board: - 3 rectangles of dimensions \(1 \times 1\) (all squares) - 2 rectangles of dimensions \(1 \times 2\) - 1 rectangle of dimensions \(1 \times 3\) So, there are a total of 6 rectangles on a \(1 \times 3\) board.

Step 2: Finding the pattern

Observe that the number of rectangles that can be formed on a \(1 \times n\) board can be represented as follows: - number of \(1 \times 1\) rectangles = n (because there are n squares) - number of \(1 \times 2\) rectangles = n - 1 (because we can slide a \(1 \times 2\) rectangle in n - 1 positions) - number of \(1 \times 3\) rectangles = n - 2 (similarly to the \(1 \times 2\) case) - ... - number of \(1 \times n\) rectangles = 1 (since there is only one possible full row)

Step 3: Formulating the recurrence relation

From the pattern above, we can see that at each stage, the total number of new rectangles added is n, where n is the dimension of the rectangular board. We can now write the recurrence relation as: \[a_n = a_{n-1} + n\] With the base case: \(a_1 = 1\) This recurrence relation can now be used to calculate the number of rectangles that can be formed on any given \(1 \times n\) rectangular board.

Key Concepts

Rectangular Board

A rectangular board in this context refers to a long strip that is 1 unit wide and extends to 'n' units in length. This configuration simplifies the task of counting rectangles since there is minimal variation in one of the dimensions.

Understanding the board as a linear extension helps in breaking down the problem. The positions along this single row allow us to easily compute different rectangles, which are essentially line segments of the strip. Knowing that each segment can be a rectangle provides a foundation for determining all possible combinations and configurations as 'n' increases.

Number of Rectangles

The number of rectangles on a rectangular board is determined by counting all possible rectangles that can be formed including squares which are a special type of rectangle. On a 1x1 board, there is just one obvious rectangle-the entire board itself.

For larger boards, this means:

• Each square or segment along the board length can be a unique rectangle.
• We count rectangles of different lengths stretching across this board.

For example, a 1x2 board has three possible rectangles: two 1x1 rectangles and one 1x2 rectangle.
This principle extends as the board length increases, showing how crucial position and overlap considerations are in determining the total count.

Pattern Observation

Observing patterns is a key part of solving recurrence relations. By examining smaller cases, patterns emerge that allow predictions for larger cases.

Each time we increase the board's length by one, we add a number of new rectangles that equals the board's new length. For a board of length **n**, rectangles formed are:

• 'n' rectangles of size 1x1
• 'n-1' rectangles of size 1x2
• 'n-2' rectangles of size 1x3
• ... up to 1 rectangle of size 1xn

This series of observations not only creates a visual pattern but also a mathematical one that can be expressed through a recurrence relation.
It highlights why observing smaller cases is essential for forming general rules.

Base Case

In calculating recurrence relations, the base case is the simplest form from which other outcomes can be determined. For our rectangles, the base case begins with the smallest board, a 1x1 rectangle.

With only one possible rectangle, this base case is simple but pivotal. It serves as a point of reference for calculating rectangles on larger boards.
From here, each increment in board length builds on previous calculations, using the base case to ensure continuity and correctness in more complex scenarios.
By establishing a solid starting point, the base case helps us develop a reliable recurrence relation that reflects the pattern across all board sizes.