Problem 42 - Precalculus with Limits

Question

In Exercises 41 - 46, write the first five terms of the arithmetic sequence defined recursively. \( a_1 = 6, a_{n + 1} = a_n + 5 \)

Step-by-Step Solution

Answer

The first five terms of the sequence are: 6, 11, 16, 21, 26.

Step 1: Understand the given recursive formula

The given formula \( a_1 = 6, a_{n + 1} = a_n + 5 \) is a recursive definition of an arithmetic sequence. The first term \( a_1 \) is 6, and each subsequent term \( a_{n + 1} \) is the sum of the previous term \( a_n \) and 5.

Step 2: Calculate the second term

Replace \( n \) with 1 in the formula \( a_{n + 1} = a_n + 5 \) to get \( a_{1 + 1} = a_1 + 5 \). This simplifies to \( a_2 = 6 + 5 = 11 \).

Step 3: Calculate the third term

Replace \( n \) with 2 in the formula \( a_{n + 1} = a_n + 5 \) to get \( a_{2 + 1} = a_2 + 5 \). This simplifies to \( a_3 = 11 + 5 = 16 \).

Step 4: Calculate the fourth term

Replace \( n \) with 3 in the formula \( a_{n + 1} = a_n + 5 \) to get \( a_{3 + 1} = a_3 + 5 \). This simplifies to \( a_4 = 16 + 5 = 21 \).

Step 5: Calculate the fifth term

Replace \( n \) with 4 in the formula \( a_{n + 1} = a_n + 5 \) to get \( a_{4 + 1} = a_4 + 5 \). This simplifies to \( a_5 = 21 + 5 = 26 \).

Key Concepts

Recursive Formula

A recursive formula is a way to define a sequence where each term is expressed in relation to the previous one, rather than as an independent expression. In our example, we start with the first term given as \( a_1 = 6 \). Each new term, \( a_{n+1} \), is calculated by adding a fixed number (here, 5) to the preceding term, \( a_n \).
This approach is like climbing stairs where each step relies on the height reached by the previous step. The recursive formula tells us how far to step upwards, and the initial term shows us where to start.
Recursive formulas are very efficient for calculations where the order of sequence matters, but require each term to be known to calculate the next.

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers where each term after the first is derived by adding a constant, known as the common difference, to the previous term. This "difference" remains consistent throughout the sequence.
In our case, the common difference is 5, meaning we add 5 to each term to find the next one. The sequence starts with \( a_1 = 6 \), and we keep adding 5 for subsequent terms, resulting in the series \( 6, 11, 16, 21, 26, \ldots \).
To identify an arithmetic progression, look for the presence of a regular adding pattern. Understanding this pattern helps in predicting the next numbers or terms in any such sequence.

Sequence Terms

The terms of a sequence are essentially the individual elements or numbers in a list. In any sequence, particularly in arithmetic ones, these terms are generated from a rule or pattern.
Think of them as numbered positions in a line, where each position holds a specific value. For the sequence we are examining, the terms can be enumerated as follows: \( a_1 = 6 \), \( a_2 = 11 \), \( a_3 = 16 \), \( a_4 = 21 \), and \( a_5 = 26 \). Each term builds on the previous one based on the rule of adding 5.
Analyzing sequence terms allows us to understand the composition and order, making it easier to establish patterns, predict future entries, and apply them in equations or real-world problems.