Question

Read the following algorithm: 10 Let \(\mathrm{P}=1\), Sum \(=0\) 20 If \(\mathrm{P} \leq \mathrm{N}\), then \((\mathrm{N}\) is the given Number) 30 Sum \(=\) Sum \(+\mathrm{P}\) 40 \(\mathrm{P}=\mathrm{P}+1\) 50 Repeat this loop 60 End Above algorithm is used to ______. (1) find first \(\mathrm{N}\) natural numbers (2) find sum of first \(\mathrm{N}\) natural numbers (3) find the sum of first \(\mathrm{N}\) even numbers (4) find the sum of first \(\mathrm{N}\) odd numbers

Short answer

Answer: The purpose of the given algorithm is to find the sum of first N natural numbers.

Step-by-step solution

Initialize Variables

First, P is set to 1 and Sum is set to 0. This means we will start with P as 1, and Sum as the accumulator of the sum of numbers.

Check Condition

The algorithm checks if P is less than or equal to the given number N. If the condition is satisfied, the loop will proceed to the next step.

Update Sum

If P is less than or equal to N, Sum is updated by adding the current value of P to it. In this step, we are summing the values of P successively as we go through the loop.

Increment P

The value of P is incremented by 1. In other words, P is increased sequentially in each iteration, going from 1 to 2, 3, 4, and so on, until P becomes greater than N.

Repeat Loop

The loop will start again from Step 2 checking if the condition P<=N is satisfied, and the steps will be executed until P becomes greater than N. Then, the loop will end.

End of Algorithm

The algorithm will finish once the loop ends, and the final value of the Sum variable will be a result of the algorithm. Now, let's determine which option fits the purpose of this algorithm: - (1) find first N natural numbers: The algorithm eventually adds the natural numbers but doesn't only find them. - (2) find sum of first N natural numbers: This option is correct since the algorithm sums all the natural numbers from 1 up to N. - (3) find the sum of first N even numbers: The algorithm doesn't only sum even numbers; it includes odd numbers as well. - (4) find the sum of first N odd numbers: Similarly, the algorithm doesn't only sum odd numbers; it includes even numbers as well. Based on the step-by-step analysis, the purpose of the given algorithm is (2) find sum of first N natural numbers.

Key concepts

Iterative Algorithms

Iterative algorithms are fundamental in programming and are designed to repeat a process until a specific condition is met. In the context of summing natural numbers, an iterative algorithm can be constructed to add successive integers, starting from 1 and continuing until a given number, which we'll call N.

At each iteration, the current number is added to an accumulator variable that holds the ongoing total. The algorithm terminates once the specified condition—the current number exceeding N—is reached. This approach is advantageous for its simplicity and is often used to demonstrate the basic principles of loops in programming education.

Sum of Natural Numbers

The sum of natural numbers can be calculated using an arithmetic progression formula or by applying an algorithmic approach. In the exercise, the latter is demonstrated. The provided algorithm outlines an iterative process for arriving at the total sum, which is a classic example of how a simple problem in mathematics can be translated into a programmatic sequence of operations.

It is essential to comprehend that the sum of the first N natural numbers is represented by the equation \( \frac{N(N + 1)}{2} \), which is derived from the series 1 + 2 + 3 + ... + N. This closed-form can be a helpful check for the correctness of our iterative solution.

Programming in Mathematics

Programming plays an increasingly crucial role in understanding and illustrating mathematical concepts. Algorithms like the one provided not only allow learners to compute values but also to grasp the logical underpinnings of mathematical operations.

Incorporating programming into mathematics education can clarify abstract ideas through concrete examples. Writing and debugging code can also enhance problem-solving skills and introduce students to computational thinking, which is invaluable in many scientific and engineering fields.

Algorithms in Education

The study of algorithms is at the heart of computer science education but can also be instrumental across various academic subjects. The iterative algorithm to sum natural numbers is an excellent teaching tool, as it embodies fundamental programming constructs such as initialization, conditionals, loops, and increments.

By breaking down this process, students learn the importance of precise instruction sequencing and control flow in problem-solving. Furthermore, educators can build on these concepts to introduce more complex topics like efficiency and optimization, identifying the trade-offs between different algorithmic approaches in terms of resources and time.