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Chapter 5: Problem 8

Give a recursive algorithm for finding the sum of the first n positive integers.

Short Answer

Expert verified
Use a recursive function where sum(n) = n + sum(n-1) with base case sum(1) = 1.

Step by step solution

01

Understand the Problem

To find the sum of the first n positive integers, we need to calculate the sum of all numbers from 1 to n. This problem can be solved using a recursive approach, where the solution for n depends on the solution for n-1.
02

Define the Base Case

In any recursive algorithm, defining a base case is essential to avoid infinite recursion. For this problem, the base case occurs when n is 1. The sum of the first positive integer (1) is simply 1. Hence, the base case is: sum(1) = 1.
03

Define the Recursive Step

For any positive integer n greater than 1, the sum of the first n positive integers can be obtained by adding n to the sum of the first n-1 positive integers. Mathematically, this can be represented as: sum(n) = n + sum(n-1).
04

Write the Recursive Algorithm

Combine the base case and the recursive step to write the algorithm: ```python def sum(n): if n == 1: return 1 else: return n + sum(n-1) ```This algorithm will return the sum of the first n positive integers.
05

Test the Algorithm

To ensure the algorithm works, test it with several values of n. For example: ```python print(sum(5)) # Output: 15 print(sum(10)) # Output: 55 ```These values match the expected sums, confirming the algorithm's correctness.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

recursion
Recursion is an essential concept in computer science and mathematics. It involves solving a problem by breaking it into smaller, more manageable sub-problems. In a recursive process, a function calls itself with modified arguments until it reaches a base case.
In this context, we're finding the sum of the first n positive integers. We can break this problem into the sum of the first n-1 integers plus n. The function calls itself repeatedly with decreasing values of n until it reaches the base case of 1.
Key points about recursion:
  • It must have a base case to stop the recursion.
  • Each recursive call should bring it closer to the base case.
  • Recursion helps avoid complex loops and makes solutions more elegant.
Let's connect it to our case: sum(n) = n + sum(n-1), where sum(1) = 1.
algorithm design
Algorithm design is the process of creating a step-by-step procedure to solve a given problem. A well-designed algorithm is efficient and easy to understand, debug, and maintain.
To design a recursive algorithm for finding the sum of the first n positive integers, we need to:
  • Understand the problem thoroughly.
  • Define a base case to stop the recursion.
  • Determine the recursive step that breaks the problem into smaller instances.
In our example:
- The problem is to find the sum of integers from 1 to n.
- The base case is when n equals 1.
- The recursive step involves calculating sum(n) = n + sum(n-1).
This design ensures our algorithm effectively tackles the problem in a structured way.
base case
A base case is a condition that stops the recursion. Without a base case, the function would call itself indefinitely, leading to an infinite loop.
In our recursive algorithm for finding the sum of the first n positive integers, the base case is when n equals 1.
For instance: if n = 1, then sum(1) = 1. This condition ensures that the recursion halts, providing a concrete result.
Key characteristics of a base case:
  • It provides the simplest, smallest instance of the problem.
  • It prevents infinite recursion.
  • It is essential for the proper functioning of a recursive algorithm.
In our algorithm, the base case gives us the sum directly without needing further recursion.
mathematical induction
Mathematical induction is a method of proof in mathematics used to establish that a given statement holds true for all natural numbers. Though not directly involved in coding a recursive algorithm, understanding it deepens our grasp on why the algorithm works.
The process involves two steps:
  • Base Case: Proving the statement is true for the initial value (usually 1).
  • Inductive Step: Assuming the statement holds for some arbitrary natural number k, and then proving it also holds for k+1.

Let's apply this to our example.
We've already set our base case: sum(1) = 1.
For the inductive step, assume sum(k) = k + sum(k-1) holds true. We need to prove sum(k+1) = (k+1) + sum(k).
By our assumption, sum(k) = k + sum(k-1), so sum(k+1) = (k+1) + [k + sum(k-1)]. This confirms recursive algorithms work through strong mathematical foundations, ensuring reliability.

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Most popular questions from this chapter

Give a recursive definition of each of these sets of ordered pairs of positive integers. \([\text {Hint} : \text { Plot the points in the set in }\) the plane and look for lines containing points in the set. \(]\) a) \(S=\left\\{(a, b) | a \in \mathbf{Z}^{+}, b \in \mathbf{Z}^{+}, \text { and } a+b \text { is odd }\right\\}\) b) \(S=\left\\{(a, b) | a \in \mathbf{Z}^{+}, b \in \mathbf{Z}^{+}, \text { and } a | b\right\\}\) c) \(S=\left\\{(a, b) | a \in \mathbf{Z}^{+}, b \in \mathbf{Z}^{+}, \text { and } 3 | a+b\right\\}\)

Give a recursive definition of the set of bit strings that are palindromes.

Give a recursive definition of \(w^{i},\) where \(w\) is a string and \(i\) is a nonnegative integer. (Here \(w^{i}\) represents the concatenation of \(i\) copies of the string \(w . )\)

A guest at a party is a celebrity if this person is known by every other guest, but knows none of them. There is at most one celebrity at a party, for if there were two, they would know each other. A particular party may have no celebrity. Your assignment is to find the celebrity, if one exists, at a party, by asking only one type of question asking a guest whether they know a second guest. Everyone must answer your questions truthfully. That is, if Alice and Bob are two people at the party, you can ask Alice whether she knows Bob; she must answer correctly. Use mathematical induction to show that if there are \(n\) people at the party, then you can find the celebrity, if there is one, with 3\((n-1)\) questions. [Hint: First ask a question to eliminate one person as a celebrity. Then use the inductive hypothesis to identify a potential celebrity. Finally, ask two more questions to determine whether that person is actually a celebrity. \(]\)

Use the principle of mathematical induction to show that \(P(n)\) is true for \(n=b, b+1, b+2, \ldots,\) where \(b\) is an integer, if \(P(b)\) is true and the conditional statement \(P(k) \rightarrow\) \(P(k+1)\) is true for all integers \(k\) with \(k \geq b\) .
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