Problem 71 - Algebra 2

Question

Write an explicit and a recursive formula for each arithmetic sequence. $$ -3,0,3,6, \dots $$

Step-by-Step Solution

Answer

For the arithmetic sequence \(-3, 0, 3, 6, \dots\), the explicit formula is \(a_n = 3n - 6\), and the recursive formula is \(a_n = a_{n-1} + 3\) with the first term \(a_1 = -3\).

Step 1: Determine the Common Difference

In order to determine the common difference in the arithmetic sequence, select any term in the sequence and subtract the preceding term. For instance, when \(0 - (-3)\) is computed, the result is \(3\). Therefore, the common difference, \(d\), in this arithmetic sequence is \(3\).

Step 2: Write the Explicit (Closed Form) Formula

The explicit formula for an arithmetic sequence can be calculated using the formula: \(a_n = a_1 + (n - 1) * d\), where \(a_n\) represents the \(n^{th}\) term in the sequence, \(a_1\) denotes the first term, \(n\) signifies the number of terms, and \(d\) stands for the common difference. By inserting the values found for \(a_1 = -3\) and \(d = 3\), the explicit formula becomes \(a_n = -3 + (n - 1) * 3.\) After simplifying, this turns into \(a_n = 3n - 6\).

Step 3: Write the Recursive Formula

The recursive formula for an arithmetic sequence can be represented in the form: \(a_n = a_{n-1} + d\), where \(a_{n}\) is the \(n^{th}\) term, \(a_{n-1}\) is the previous term, and \(d\) represents the common difference. In this case, \(d = 3\), and the first term, \(a_1 = -3\). Therefore, the recursive formula for this sequence is \(a_n = a_{n-1} + 3\), with \(a_1 = -3\).

Key Concepts

Explicit Formula

An explicit formula in the context of an arithmetic sequence allows you to find any term in the sequence without needing to know the preceding one. This is like having a direct map to every part of your journey through the sequence!
The explicit formula for an arithmetic sequence is represented as

• \(a_n = a_1 + (n - 1) \cdot d\)

where:

• \(a_n\) is the \(n^{th}\) term you are trying to find,
• \(a_1\) is the first term of the sequence,
• \(d\) is the common difference between consecutive terms,
• and \(n\) is the position of the term within the sequence.

For the sequence given:

• The first term \(a_1 = -3\),
• and the common difference \(d = 3\).

Substituting these into the formula, you get:

• \(a_n = -3 + (n - 1) \cdot 3\)

which simplifies to \(a_n = 3n - 6\). This formula now acts as a quick solution to determine any term in the sequence just by plugging in the term position \(n\). You don’t need the previous terms to find your desired term.

Recursive Formula

The recursive formula, unlike the explicit formula, requires you to know the previous term in order to find the next one. It’s akin to stepping stone by stone along your journey through the sequence.
The recursive formula for an arithmetic sequence is structured as:

• \(a_n = a_{n-1} + d\),

where:

• \(a_n\) is the \(n^{th}\) term,
• \(a_{n-1}\) is the term before it,
• and \(d\) is the common difference.

For our example sequence:

• \(d = 3\),
• and \(a_1 = -3\).

The recursive formula becomes:

• \(a_n = a_{n-1} + 3\)

This relationship establishes the need to calculate each term based on its predecessor, starting from the initial term \(a_1 = -3\). In recursive sequences, you essentially build the sequence term by term.

Common Difference

In any arithmetic sequence, the common difference is a crucial concept. It is the consistent gap from one term to the next, and it guides how the sequence grows or decreases.
To find the common difference, select any term and subtract the term before it. For instance, looking at the sequence

• -3, 0, 3, 6, ...

and calculating:

• \(0 - (-3) = 3\),
• \(3 - 0 = 3\),
• \(6 - 3 = 3\)

It verifies that each term is consistently increasing by 3.
This uniformity in growth is the essence of an arithmetic sequence, making the sequence predictable. Knowing the common difference \(d\) helps you write both the explicit and recursive formulas for an arithmetic sequence. This difference, when added repeatedly, spans across all the terms, defining their order and value.