Problem 54 - Algebra 2

Question

Write an explicit and a recursive formula for each sequence. \(-5,-4,-3,-2,-1, \ldots\)

Step-by-Step Solution

Answer

The recursive formula of the sequence \(-5,-4,-3,-2,-1, \ldots\) is \(a_1 = -5\) and \(a_n = a_{(n-1)} + 1\) for \(n > 1\), and the explicit formula is \(a_n = -6 + n\).

Step 1: Determine the common difference

Look at the sequence \(-5,-4,-3,-2,-1, \ldots\) and subtract the second term by the first term to determine the common difference: \(-4-(-5)=1\). Thus, the common difference, denoted as 'd', is 1.

Step 2: Write the recursive formula

A recursive formula for a sequence specifies how some of the terms after the first term are calculated from the preceding terms. For an arithmetic sequence, each subsequent term can be found by adding the common difference to the previous term. Our first term (\(a_1\)) is -5 and the common difference (d) is 1, so the recursive formula of the sequence will be: \(a_1 = -5\) and \(a_n = a_{(n-1)} + 1\) for \(n > 1\). This means the first term is -5, and any term after the first is found by adding 1 to the previous term.

Step 3: Write the explicit formula

An explicit formula for an arithmetic sequence is in the form \(a_n = a_1 + (n-1) * d\), where \(a_n\) is the nth term, \(a_1\) is the first term, and d is the common difference. Here, our first term (\(a_1\)) is -5 and the common difference (d) is 1, so the explicit formula of the sequence will be \(a_n = -5 + (n-1)*1 = -6 + n\). This formula allows us to calculate the nth term of the sequence directly without knowing the previous terms.

Key Concepts

Recursive Formula

A recursive formula helps us generate a sequence by relating each term to its predecessor. Think of it as a step-by-step process where each term is built upon the last. For an arithmetic sequence, this involves adding a constant value, known as the common difference, to get from one term to the next.

In the provided sequence \(-5, -4, -3, -2, -1, \ldots\), the recursive formula starts with the initial term known as the first term, defined as \(a_1 = -5\). Then, each subsequent term \(a_n\) can be found by the relation \(a_n = a_{(n-1)} + 1\) for \(n > 1\). Here, "1" is the common difference.

This formula effectively means you start from \(-5\) and for every next term, just add 1 to the previous term. It’s like climbing steps on a ladder one at a time. This method is especially useful if you want to find the next few terms in a sequence without extensive calculations.

Explicit Formula

An explicit formula is like having a magic key that opens any lock in one move. Instead of calculating each term from the previous one, an explicit formula tells you directly how to find any term in the sequence.

For arithmetic sequences, the explicit formula is given by \(a_n = a_1 + (n-1) \cdot d\). Here, \(a_n\) is the term you wish to find, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

In this problem, we use \(-5\) as \(a_1\) and "1" as \(d\), forming the explicit formula: \(a_n = -5 + (n-1) \cdot 1\). Simplifying this gives \(a_n = n - 6\).

Using this formula, if you want to find the 10th term, just substitute \(n = 10\) into the formula, and effortlessly find your term without any intermediate steps. Explicit formulas are especially convenient when needing a specific term's value far into the sequence.

Common Difference

The term "common difference" is central to understanding arithmetic sequences. It is the constant amount added or subtracted to each term to reach the next one in the sequence. This value is consistent throughout the sequence, meaning each term increases or decreases uniformly.

To determine it, you subtract any term from the subsequent term. In our sequence \(-5, -4, -3, -2, -1, \ldots\), by subtracting the second term from the first (or any other two consecutive terms), we have: \(-4 - (-5) = 1\). Hence, the common difference here is \(1\).

This common difference "1" tells you exactly how much you need to move from one term to the next. It's like knowing the steps in a dance that repeat identically. Recognizing the common difference allows you to craft both recursive and explicit formulas for the sequence, giving you two tools to find any term systematically.