Problem 57 - Discrete Mathematics with Applications

Question

Let \(a_{n}\) denote the number of times the assignment statement \(x \leftarrow x+1\) is executed by each nested for loop. Define \(a_{n}\) recursively. $$ \begin{array}{c}{\text { for } 1=1 \text { to } n \text { do }} \\ {\text { for } j=1 \text { to } i \text { do }} \\ {x \leftarrow x+1}\end{array} $$

Step-by-Step Solution

Answer

The recursive formula for \(a_n\) representing the number of times the assignment statement \(x \leftarrow x+1\) is executed by the nested loop is: $$ a_n = \begin{cases} 1 & n=1\\ a_{n-1} + n & n > 1 \end{cases} $$

Step 1: Observe the loop structure

The outer loop runs from 1 to n, which means it will execute n times. In each iteration of the outer loop, the inner loop runs from 1 to i. For example, when i=1, the inner loop will run once; when i=2, the inner loop will run twice, and so on. In general, for any i, the inner loop will execute i times.

Step 2: Calculate the number of times the statement is executed for each iteration of the outer loop

When i=1, the inner loop will run once, so the statement will be executed once: \(a_1 = 1\). When i=2, the inner loop will run twice, so the statement will be executed twice: \(a_2 = 2\). When i=3, the inner loop will run three times, so the statement will be executed three times: \(a_3 = 3\). In general, for any i, the statement will be executed i times: \(a_i = i\).

Step 3: Define the recursive formula for \(a_n\)

From the previous observations, we can deduce that the assignment statement \(x \leftarrow x+1\) is executed \(a_i = i\) times for any i. To find the total number of times the statement is executed for n iterations of the outer loop, we can sum up the individual counts from \(a_1\) to \(a_n\): $$a_n = a_{n-1} + n$$ With the base case: $$a_1 = 1$$ So the recursive formula for \(a_n\) is: $$ a_n = \begin{cases} 1 & n=1\\ a_{n-1} + n & n > 1 \end{cases} $$

Key Concepts

nested loops

Nested loops are an essential programming construct where one loop is placed inside another. In our example, we have a nested loop structure with an outer loop running from 1 to \( n \) and an inner loop running from 1 to \( i \), where \( i \) is the current iteration count of the outer loop.

Nested loops are powerful because they allow multiple levels of iteration, often used to traverse multi-dimensional data structures or solve complex mathematical problems.

• The outer loop dictates how many times the entire process will be repeated.
• The inner loop defines how many times a particular action is performed during each of these repetitions.

In this exercise, each time the outer loop executes, the inner loop contributes to incrementing a variable \( x \) a certain number of times, based on the current outer loop iteration. This repetitive behavior is the hallmark of nested loops.

recursive formula

Recursive formulas are equations where each term is defined as a function of its preceding terms. In this scenario, the assignment statement \( x \leftarrow x+1 \) is repeated in a structured manner described by our nested loops.

Here, we define the number of times this increment happens through a recursive formula:

• For the first iteration, the increment is straightforward: \( a_1 = 1 \).
• For subsequent iterations, the total executions include all previous executions plus the current iteration count, expressed as \( a_n = a_{n-1} + n \).

This form of recursive definition captures not just the immediate number of incremental actions but also builds upon the cumulative effect seen in previous steps, making it ideal for summing sequences as encountered in these loops.

base case

In recursive functions, a base case serves as a foundation or stopping point that does not further deconstruct into smaller problems. It is crucial for preventing infinite recursions.

For our problem, the base case is \( a_1 = 1 \). This initial step is the simplest form of the problem, whereby with only one iteration in the outer loop, the inner loop also runs just once.

• The base case provides the simplest scenario, with no additional dependencies.
• It ensures that the recursion has an endpoint, thereby allowing the recursive formula \( a_n = a_{n-1} + n \) to effectively compute values for any \( n \).

By defining a clear base case, we guarantee that our recursive definition will correctly handle the first instance and build logically from there without running endlessly.

loop iterations

Understanding loop iterations is key to grasping how operations are repeated in programming. Loop iterations describe how many times a block of code will be executed.

In our example, the outer loop iterates from 1 to \( n \). The number of iterations indicates how many times the inner loop gets executed for each specific \( i \).

• Each iteration of the outer loop calls for the inner loop to execute a corresponding number of times.
• The cumulative effect of multiple iterations results in the total number of increments performed by the nested loops.

Consider each loop iteration as a cycle in which the inner loop's execution is dependent on the outer iteration count, compounding the number of times \( x \) is incremented. Understanding this structure helps in foreseeing how complex operations could be broken down or built up via iterative processes.