Problem 41 - Precalculus with Limits

Question

In Exercises 41 - 46, write the first five terms of the arithmetic sequence defined recursively. \( a_1 = 15, a_{n + 1} = a_n + 4 \)

Step-by-Step Solution

Answer

The first five terms of the sequence are 15, 19, 23, 27, 31.

Step 1: Find the First Term

The first term is given as \(a_1 = 15\). So the first term of the sequence is 15.

Step 2: Find the Second Term

Use the recursive form of the arithmetic sequence to calculate the second term: \(a_{n + 1} = a_n + 4\). Substitute n=1 to find \(a_2 = a_1 + 4 = 15 + 4 = 19\)

Step 3: Find the Third Term

Repeat the process for the third term: Substitute n=2 to find \(a_3 = a_2 + 4 = 19 + 4 = 23\)

Step 4: Find the Fourth Term

Continue the pattern: Substitute n=3 to find \(a_4 = a_3 + 4 = 23 + 4 = 27\)

Step 5: Find the Fifth Term

Finally, substituting n=4 to find the fifth term: \(a_5 = a_4 + 4 = 27 + 4 = 31\)

Key Concepts

Recursive Sequences

In mathematics, sequences are lists of numbers that follow a specific pattern or rule. One type of sequence is the recursive sequence. A recursive sequence relies on the previous term to find the next term. This means that after knowing the starting term, each subsequent term is computed using the term before it.
When dealing with recursive sequences, you must be familiar with how to use the given first term and the recursive formula. In the exercise, the first term (\(a_1\)) is 15, and the recursive formula is \(a_{n+1} = a_n + 4\). Here, the formula tells us how to find the next term by adding 4 to the current term.

• Start with the initial term: \(a_1 = 15\)
• Apply the recursive rule to find subsequent terms.

Thus, recursive sequences provide a systematic approach to building a sequence from one term to the next.

Sequence Terms

Sequence terms are the individual elements or values in the sequence. They can be considered as the "steps" in the arithmetic journey we are on when following a sequence. Each term depends on its position, and the way each term is related to others sets the sequence's pattern.
In the exercise, the first five sequence terms are calculated using the recursive method:

• Starting from the given: \(a_1 = 15\)
• The second term: \(a_2 = 19\)
• The third term: \(a_3 = 23\)
• The fourth term: \(a_4 = 27\)
• The fifth term: \(a_5 = 31\)

By computing each term from the previous one, we build the sequence step by step. It's like laying bricks one after another to build a pathway.

Common Difference

In arithmetic sequences, the common difference is a crucial element. It tells us how much each term increases by from the previous one. This constant difference between consecutive terms makes the sequence arithmetic.
In the given exercise, the common difference is 4, which is evident in the recursive formula \(a_{n+1} = a_n + 4\).

• Each term increases by 4 over the previous term.
• This constant addition makes each term farther down the number line by a set amount.

Understanding the common difference is like knowing the regular steps in your sequence journey. It provides predictability, which is essential in solving or generating terms of the sequence.