Problem 20 - Precalculus: Mathematics for Calculus

Question

A sequence is defined recursively by the given formulas. Find the first five terms of the sequence. \(a_{n}=a_{n-1}+a_{n-2}+a_{n-3} \quad\) and \(\quad a_{1}=a_{2}=a_{3}=1\)

Step-by-Step Solution

Answer

The first five terms are: 1, 1, 1, 3, 5.

Step 1: Understand the Recursive Formula

The sequence is defined recursively, which means that each term depends on previous terms in the sequence. The formula given is \(a_{n} = a_{n-1} + a_{n-2} + a_{n-3}\). The initial terms are \(a_{1} = a_{2} = a_{3} = 1\). This means the first three terms are already provided: 1, 1, and 1.

Step 2: Calculate the Fourth Term

Using the recursive formula, calculate \(a_4\) by substituting the known terms: \(a_4 = a_3 + a_2 + a_1\). Since \(a_3 = 1\), \(a_2 = 1\), and \(a_1 = 1\), we have: \(a_4 = 1 + 1 + 1 = 3\).

Step 3: Calculate the Fifth Term

Now, calculate \(a_5\) using the recursive formula and the most recent terms: \(a_5 = a_4 + a_3 + a_2\). We know \(a_4 = 3\), \(a_3 = 1\), and \(a_2 = 1\), so \(a_5 = 3 + 1 + 1 = 5\).

Step 4: Review the First Five Terms

We've calculated the first five terms of the sequence as follows: \(a_1 = 1\), \(a_2 = 1\), \(a_3 = 1\), \(a_4 = 3\), and \(a_5 = 5\).

Key Concepts

Sequence Terms

A sequence is essentially a list of numbers where each number is referred to as a term. In recursive sequences, such as the one in the exercise, each new term is determined by a rule, called a recursive formula, which uses preceding terms to find the next one. For instance, in our sequence, the first few terms are defined explicitly and used to find subsequent terms.

• First term (\(a_1\)): 1
• Second term (\(a_2\)): 1
• Third term (\(a_3\)): 1
• Fourth term (\(a_4\)): Derived using the recursive formula
• Fifth term (\(a_5\)): Derived using the recursive formula as well

This list grows by continuously applying the relationship defined in the recursive formula. It's like following a blueprint where each step depends directly on the ones that came before.

Initial Conditions

Initial conditions in a recursive sequence are like the starting point or seed that kicks off the sequence. These are the specific values given for the first few terms, and they are crucial because any subsequent terms are computed based on them. In this exercise, the initial conditions were clearly specified as \(a_1 = a_2 = a_3 = 1\).

These values ensure that when we use the recursive formula, there are no objections or uncertainties—the sequence has a reliable foundation to build upon. Without these initial conditions, defining the sequence with only a recursive rule would be incomplete and ambiguous. Think of initial conditions as the launching pad for discovering the pattern or progression of the entire sequence.

Recursive Formula

The recursive formula is the heart of any recursive sequence, acting as a command that tells us how to compute subsequent terms based on earlier terms, like a mathematical chain reaction. In our specific sequence, the recursive formula is given by \(a_{n} = a_{n-1} + a_{n-2} + a_{n-3}\). This notation implies that each term, starting from the fourth, is the sum of the previous three terms.

• For \(a_4\), the formula becomes: \(a_4 = a_3 + a_2 + a_1\)
• For \(a_5\), it applies as: \(a_5 = a_4 + a_3 + a_2\)

The real beauty of recursive formulas is their ability to systematically extend a sequence without needing to re-evaluate or recalculate the entire timeline. Each term is crafted using a replicable approach, offering an efficient way to find terms further down the line. Understanding the recursive formula is key to forecasting and analyzing the behavior of the sequence as it expands.