Problem 39 - Calculus Single Variable

Question

Give a recursive definition of the sequence. $$ f_{n}=n(n+1) / 2, n=1,2,3, \ldots $$

Step-by-Step Solution

Answer

The recursive definition is: Base case: \(f_1 = 1\), Recursive step: \(f_{n+1} = f_n + (n+1)\).

Step 1: Understand the Explicit Definition

The explicit formula given is \( f_n = \frac{n(n+1)}{2} \). This is a formula for the \(n\)-th term of the sequence, which calculates the sum of the first \(n\) natural numbers. Our task is to convert this into a recursive definition.

Step 2: Analyze the Base Case of the Sequence

For any recursive definition, we need a base case. Let's find \(f_1\) using the explicit formula. \(f_1 = \frac{1(1+1)}{2} = 1\). Thus, the base case is \(f_1 = 1\).

Step 3: Identify the Recursive Relationship

Observe that \(f_n\) is the sum of the first \(n\) numbers. Therefore, to get \(f_{n+1}\), we add \(n+1\) to \(f_n\). Hence, the recursive relation is \(f_{n+1} = f_n + (n+1)\).

Step 4: Verify the Recursive Formula

Check the recursive formula for a few terms. If \(f_1 = 1\), then \(f_2 = f_1 + 2 = 1+2 = 3\). Continuing this, \(f_3 = f_2 + 3 = 3+3 = 6\), which matches the explicit formula \(\frac{3(3+1)}{2} = 6\). The recursive relation works.

Key Concepts

Explicit Formula

An explicit formula gives a direct calculation for finding the term in a sequence. With the explicit formula, you don't need to know prior terms to calculate the next term. For the sequence described in the exercise, the explicit formula is given as \( f_n = \frac{n(n+1)}{2} \). This formula allows us to find the \(n\)-th term without computing the previous terms. Conversely, it helps to understand the nature of the sequence, which in this case is the sum of the first \(n\) natural numbers. With an explicit formula, one has immediate access to any sequence term through direct substitution.

Base Case

In a recursive sequence, establishing a base case is crucial because it serves as the starting point from which all other terms are generated. For the sequence provided, the base case is determined by substituting \(n = 1\) into the explicit formula. This gives us \(f_1 = \frac{1(1+1)}{2} = 1\). The base case essentially provides the first term of the sequence, ensuring that the recursive rule has a definite starting point. Without a base case, the recursive sequence would be incomplete as there would be no known value to begin the sequence.

Recursive Relationship

A recursive relationship is a way of defining a sequence based on its preceding terms. This means you express \( f_{n+1} \) as a function of \( f_n \). In our exercise, the recursive relationship is given by \( f_{n+1} = f_n + (n+1) \), which means each term is generated by adding \( n+1 \) to the previous term. Recursive relationships are powerful because they can be simpler to work with when calculating terms successively. They provide a systematic way to calculate every future value from the initial value, making them essential for constructing sequences when an explicit formula isn't available or practical.

Natural Numbers

Natural numbers are a set of positive integers starting from 1 and going onward (1, 2, 3,…). They are fundamental in mathematics as they represent the most basic form of counting. In the context of our recursive sequence, we are using natural numbers as the index \( n \), to define terms of the sequence. Natural numbers form the basis on which sequences such as arithmetic sequences (where each term after the first is obtained by adding a constant) and geometric sequences (where each term is a constant multiple of the previous one) are developed. They are essential in applying recursive formulas, as these generally iterate over natural numbers.