Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Find the first four terms of each sequence. $$a_{1}=5, a_{n}=3 n+3 a_{n-1}, \text { for } n>1$$

Step-by-Step Solution

Answer

First four terms are 5, 21, 72, 228.

Step 1: Calculate the First Term

The sequence is given such that the first term is already defined. We have:\[ a_1 = 5 \]Thus, the first term is 5.

Step 2: Calculate the Second Term

The formula for any term \( a_n \) is given by \( a_n = 3n + 3a_{n-1} \).To find \( a_2 \), substitute \( n = 2 \) and \( a_1 = 5 \) into the formula:\[a_2 = 3 imes 2 + 3 imes 5 = 6 + 15 = 21\]Thus, the second term is 21.

Step 3: Calculate the Third Term

Substitute \( n = 3 \) and \( a_2 = 21 \) into the formula \( a_n = 3n + 3a_{n-1} \):\[a_3 = 3 imes 3 + 3 imes 21 = 9 + 63 = 72\]Thus, the third term is 72.

Step 4: Calculate the Fourth Term

Substitute \( n = 4 \) and \( a_3 = 72 \) into the formula \( a_n = 3n + 3a_{n-1} \):\[a_4 = 3 imes 4 + 3 imes 72 = 12 + 216 = 228\]Thus, the fourth term is 228.

Key Concepts

Understanding Sequence Calculation

In mathematics, sequences are ordered lists of numbers where each number is called a term. Calculating the terms of a sequence can be done using various methods, depending on how the sequence is defined. A recursive sequence is one where each term is defined in relation to the previous terms.
For example, consider a sequence where the first term is given, and subsequent terms are calculated using a specific formula involving the previous term(s). Let’s break down the process for sequence calculation:

• Start with a given term; this is often called the initial condition.
• Use the recursive formula to calculate the next terms.
  Repeat this process for each subsequent term needed to meet the requirement.

Understanding the recursive relationship and how it connects each term is critical.
It involves both arithmetic and algebraic concepts, making it important to follow each step carefully to avoid calculation errors.

Concept of a Term Formula

A term formula is essential for calculating the terms of a sequence. It defines how each term relates to the term before it, using a mathematical expression. This formula forms the foundation of the process to find successive terms in a sequence.
In our example, the term formula is:\[ a_n = 3n + 3a_{n-1} \]This formula involves two parts:

• The arithmetic part: \( 3n \), where \( n \) changes as we calculate each new term.
• The recursive part: \( 3a_{n-1} \), where \( a_{n-1} \) is the previous term already known from prior calculation.

The term formula's actual implementation involves strategic substitution and algebraic simplification. It’s crucial to replace \( n \) with the correct value for each respective term, ensuring accurate results.
Being precise with terms and their calculations underscores the importance of following the formula faithfully at each step.

Executing a Step-by-Step Solution

Approaching a recursive sequence involves a step-by-step plan where each step builds upon the last. To find multiple terms, like in our example, it is crucial to be methodical:
1. **Start by identifying the first term**, simply given as part of the problem's initial condition.

• Here, \( a_1 = 5 \).

2. **Apply the term formula** for the next term by substituting values:

• Calculate \( a_2 \) by substituting \( n = 2 \) and \( a_1 = 5 \) into the formula.
• Continue calculating \( a_3 \) and \( a_4 \) by changing \( n \) to 3 and 4, respectively, using their previous terms.

3. The formula’s recursive feature links each new term to the prior one, ensuring that missing steps are avoided through correct substitutions.
The gradual build-up from one term to the next not only assures the correctness of each calculation but also highlights how interdependent each term is within the sequence.