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Chapter 5: Problem 9

Let \(F\) be the function such that \(F(n)\) is the sum of the first \(n\) positive integers. Give a recursive definition of \(F(n) .\)

Short Answer

Expert verified
F(n) = 1 if n = 1; F(n) = F(n-1) + n if n > 1

Step by step solution

01

Define base case

Identify the simplest case of the problem. For the sum of the first n positive integers, the simplest case is when n = 1. Thus, define the base case as: F(1) = 1
02

Express the recursive step

Find a relationship between F(n) and F(n-1). We know that the sum of the first n integers can be represented as the sum of the first (n-1) integers plus n. Thus, the recursive step is: F(n) = F(n-1) + n
03

Combine base case and recursive step

Combine the base case and the recursive step to define the function recursively. So, the recursive definition is: F(n) = 1, if n = 1 F(n) = F(n-1) + n, if n > 1

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Key Concepts

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

Base Case
When working with recursive functions, it's essential to define the simplest possible case, known as the 'base case.'
The base case acts as the foundation, allowing the recursion to terminate at some point.
For example, if we want to find the sum of the first n positive integers using recursion, we start by identifying the simplest scenario: when n equals 1. So, the base case is:
  • F(1) = 1
This makes sense since the sum of the first 1 positive integer is simply 1.
Without this base case, the function would keep calling itself indefinitely, leading to an infinite loop.
Thus, the base case not only simplifies the problem but also ensures the recursion stops at a certain point.
Recursive Definition
After establishing the base case, the next step is to define the function recursively.
This involves identifying a relationship between the function and its preceding values.
For the sum of the first n positive integers, think of this sum as the sum of the first (n-1) positive integers plus the next integer, n.
So, we can express this relationship as:
  • F(n) = F(n-1) + n
Here, F(n) depends on F(n-1), and we keep reducing n by 1 until we hit the base case, F(1).
This recursive definition breaks the problem into smaller sub-problems, making it easier to solve.
Such relationships help to simplify complex problems step by step through a series of recursive calls.
Sum of Integers
Combining the base case and the recursive definition gives us a complete recursive function to find the sum of the first n positive integers.
Here's how it looks:
  • If n = 1: F(1) = 1
  • If n > 1: F(n) = F(n-1) + n
For example, if you want to find the sum of the first 4 positive integers, you would follow this recursive process:
  • F(4) = F(3) + 4
  • F(3) = F(2) + 3
  • F(2) = F(1) + 2
  • F(1) = 1 (Base case)
Adding these up step-by-step, you get:
1 + 2 + 3 + 4 = 10
This recursive breakdown makes it easier to see how the sum is built up incrementally.
Using this method, you can find the sum of any number of positive integers efficiently.

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