Question

Write a recursive rule for the sequence. $$ 1,8,15,22,29, \ldots $$

Short answer

The recursive rule for the sequence 1, 8, 15, 22, 29, ... is \( a_1 = 1 \) and \( a_n = a_{n-1} + 7 \) for \( n > 1 \).

Step-by-step solution

Identify the common difference

Examine the sequence to find any common difference between consecutive terms. Going by the sequence - 1, 8, 15, 22, 29, we can see that there is a common difference of 7.

Write out the first recursive rule

The initial term of an arithmetic sequence is usually denoted as \( a_1 \). In this sequence, \( a_1 \) is 1. This is the base case of the recursive rule.

Write out the recursive rule for the rest of the sequence

Once the base case - \( a_1 \) is defined, the recursive rule for the rest of the sequence terms can be defined in relation to the previous term, adding the common difference. So, the general recursive rule will be \( a_n = a_{n-1} + 7 \), where \( a_n \) is the nth term of the sequence, and \( a_{n-1} \) is the (n-1)th term.

Key concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a specific pattern: each term after the first is derived by adding a constant value, known as the common difference, to the preceding term. This makes the sequence predictable and allows us to form a general rule to find any term in the sequence.

For instance, consider the arithmetic sequence presented in the exercise: 1, 8, 15, 22, 29,... Each number is seven more than the one before it. To describe this pattern, we need an understanding of recursive rules and the ability to identify the common difference, which is crucial for constructing such rules. Importantly, students should learn how to apply these concepts to create formulas that can predict any term in the sequence, enhancing their algebraic thinking and problem-solving skills.

Common Difference

Identifying the Common Difference

The term 'common difference' is central to understanding arithmetic sequences. It is the consistent interval between consecutive terms of the sequence. To find the common difference, simply subtract any term from the following term. For example, in the sequence from the exercise, subtracting 1 from 8, or 8 from 15, always gives us 7, which is the common difference.

Significance in Recursive Formulas

Knowing the common difference allows us to formulate recursive expressions that effectively describe how the sequence develops. For educational purposes, it helps to practice identifying the common difference by looking at various sequences, as it reinforces the concept and allows a deeper understanding of not just arithmetic sequences but algebraic thinking as well.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are the unspoken language of algebra that succinctly expresses mathematical ideas and relationships. In the context of an arithmetic sequence, the expression for a recursive rule combines a variable representing the terms in the sequence with the common difference into a formula.

In the exercise provided, for example, the recursive rule expression is formulated as \( a_n = a_{n-1} + 7 \). This expression conveniently encapsulates the entire pattern of the sequence: to get any term in the sequence, start with the preceding term and add the common difference. Through exercises like these, students become adept at translating patterns and sequences into algebraic notation, enhancing their ability to tackle more complex mathematical problems.