Question

explaining how to write the explicit rule for an arithmetic sequence

the first term of an arithmetic sequence is 42. the rule \\(a_n = a_{n-1} + 8\\) can be used to find the next term of the sequence.

explain how to write the explicit rule for the arithmetic sequence from the given information.

Explanation:

Identify the first term and common difference

Using the Arithmetic Sequences and Recursive Formulas knowledge points

$$
LATEXBLOCK0
$$

State the general explicit formula

Using the Arithmetic Sequences knowledge point

$$ a_n = a_1 + (n - 1)d $$

Substitute the known values

Using the Arithmetic Sequences knowledge point

$$ a_n = 42 + (n - 1)8 $$

Simplify the explicit rule

Using the Algebraic Simplification knowledge point

$$
LATEXBLOCK1
$$

Answer:

To write the explicit rule for the arithmetic sequence, follow these steps:

1. Identify the first term (\(a_1\)) and the common difference (\(d\)): From the given information, the first term is \(a_1 = 42\). The recursive rule \(a_n = a_{n-1} + 8\) shows that each term is found by adding \(8\) to the previous term, so the common difference is \(d = 8\).
2. Use the general explicit formula: The general formula for an arithmetic sequence is \(a_n = a_1 + (n - 1)d\).
3. Substitute the values: Replace \(a_1\) with \(42\) and \(d\) with \(8\) to get \(a_n = 42 + (n - 1)8\).
4. Simplify the expression: Distribute the \(8\) and combine like terms to get the simplified explicit rule:

$$a_n = 8n + 34$$