Problem 53 - Algebra 2

Question

Use each recursive formula to write an explicit formula for the sequence. $$ a_{1}=-5, a_{n}=a_{n-1}-1 $$

Step-by-Step Solution

Answer

The explicit formula for the given sequence is \(a_{n} = -n + 4\).

Step 1: Identify Recursive Formula Components

In the recursive formula provided, \(a_{1} = -5\) and \(a_{n} = a_{n-1} - 1\), it can be noticed that each term in the sequence is obtained by subtracting 1 from the previous term. This is a common feature of arithmetic sequences, and in those cases, the explicit formula is given by the form: \(a_{n} = a_{1} + (n - 1) \cdot d\), where \(a_{1}\) is the first term and \(d\) is the common difference between terms in the sequence. Here, the first term \(a_{1}\) is -5 and the common difference \(d\) is -1, as each term is 1 less than the previous term.

Step 2: Substitute into Explicit Formula

Substitute the known values into the formula for an arithmetic sequence so we get: \(a_{n} = -5 + (n - 1) \cdot -1\).

Step 3: Simplify the Expression

To obtain the explicit formula, simplify the expression: \(a_{n} = -5 - (n - 1) = -n - 5 + 1 = -n + 4\). So, the explicit formula for the sequence is \(a_{n} = -n + 4\). This is a formula that allows any term of the sequence to be found directly without needing to know any previous terms.

Key Concepts

Recursive Formula

A recursive formula defines the terms of a sequence using one or more previous terms. It tells us how to get from one term in the sequence to the next. This type of formula is based on specifying the first term (or a few initial terms) and then provides a rule to find each subsequent term.

In the example stated, the recursive formula is given as:

• \(a_{1} = -5\)
• \(a_{n} = a_{n-1} - 1\)

This means the sequence starts at \(-5\), and to find each subsequent term, you subtract 1 from the previous one. The recursive formula is very useful for understanding the pattern and nature of the sequence but can be cumbersome if we want to find a far-off term without going through all previous terms.

Explicit Formula

An explicit formula provides a direct way to find any term in a sequence without having to know the preceding term. This formula is beneficial because it enables us to compute any term in the sequence instantly.
The explicit formula for an arithmetic sequence is usually given by:

• \(a_{n} = a_{1} + (n - 1) \cdot d\)

where:

• \(a_{1}\) is the first term of the sequence.
• \(d\) is the common difference between consecutive terms.

For our sequence, substituting the values \( a_{1} = -5 \) and \( d = -1 \) into the formula, we obtain:

• \(a_{n} = -5 + (n - 1) \cdot (-1)\)

This simplifies to \(a_{n} = -n + 4\), allowing anyone to find the nth term directly, without following the recursive pattern.

Common Difference

The common difference is a key element of an arithmetic sequence. It is the difference between each pair of consecutive terms in the sequence. This value remains constant throughout an arithmetic sequence and is a vital part of both recursive and explicit formulas.
To find the common difference, you simply subtract any term from the subsequent term. For the sequence described in the problem, the common difference \(d\) is:

• Given as \(-1\) in the recursive formula \(a_{n} = a_{n-1} - 1\)

The use of this common difference allows us to create an explicit formula, making it easy to calculate any term in the sequence. Understanding the concept of a common difference is essential for handling arithmetic sequences effectively.