Problem 13 - Algebra 2

Question

Find the first five terms of each sequence. $$ a_{1}=6, a_{n+1}=a_{n}+n+3 $$

Step-by-Step Solution

Answer

6, 10, 15, 21, 28

Step 1: Understand the Sequence

We are given a recursive sequence where the first term \( a_1 \) is 6. The rule for the sequence is given by \( a_{n+1} = a_n + n + 3 \). This means each term is formed by adding \( n + 3 \) to the previous term.

Step 2: Calculate the Second Term

Given \( a_1 = 6 \), we can calculate \( a_2 \) using the rule: \( a_2 = a_1 + 1 + 3 = 6 + 1 + 3 = 10 \).

Step 3: Calculate the Third Term

Using the second term, \( a_2 = 10 \), calculate \( a_3 \): \( a_3 = a_2 + 2 + 3 = 10 + 2 + 3 = 15 \).

Step 4: Calculate the Fourth Term

Now, using \( a_3 = 15 \), find \( a_4 \): \( a_4 = a_3 + 3 + 3 = 15 + 3 + 3 = 21 \).

Step 5: Calculate the Fifth Term

Using \( a_4 = 21 \), determine \( a_5 \): \( a_5 = a_4 + 4 + 3 = 21 + 4 + 3 = 28 \).

Step 6: List the First Five Terms

The first five terms of the sequence are: 6, 10, 15, 21, and 28.

Key Concepts

Sequence Terms

In mathematics, a sequence refers to an ordered list of numbers where each number is called a term. A key characteristic of sequences is that terms appear in a specific order and have a defined relationship to previous terms. In the given exercise, the sequence begins with the initial term, also known as the first term, which is 6. This starting point is crucial as it sets the stage for generating the rest of the terms. Each term after the first is determined by a rule or formula that often relates to the one or more previous terms.
Recursive sequences, like the one in the exercise, are sequences where each term is defined based on the preceding terms. Here, the formula provided helps us find each successive term by continuously building upon previous ones. Understanding the structure and order of sequence terms is essential for solving any problems related to sequences.

Arithmetic Rules

Arithmetic rules in sequences help us understand how each term is generated. For recursive sequences, the arithmetic operations often involve addition or subtraction applied to terms. In this specific sequence, the rule is outlined as follows:

• Start with the first term, which is given.
• Each subsequent term is formed by adding a particular arithmetic expression to the previous term.

The exercise states the rule as \( a_{n+1} = a_n + n + 3 \). This equation signifies that to find the next term, we take the current term \( a_n \), add \( n \), the index of that term minus 1, and then add 3. It's a systematic approach that ensures the sequence progresses smoothly from term to term.
Understanding these rules is vital as they govern the entire sequence, allowing us to predict and calculate beyond just the initial terms.

Term Calculation

Calculating terms in a sequence, especially a recursive one, involves directly applying the arithmetic rule repeatedly. To derive each term in the exercise, we follow a step-by-step process:

• Begin with the initial term \( a_1 = 6 \).
• To find \( a_2 \), use the rule \( a_2 = a_1 + 1 + 3 = 10 \).
• For \( a_3 \), apply \( a_3 = a_2 + 2 + 3 = 15 \).
• The fourth term is \( a_4 = a_3 + 3 + 3 = 21 \).
• Finally, the fifth term is \( a_5 = a_4 + 4 + 3 = 28 \).

Each step builds on the last, effectively generating the sequence. It's a process that highlights the recursive nature of the sequence, showing how each term depends on its predecessor. With practice, recognizing and applying these patterns becomes second nature, allowing for the generation of even more terms if needed.