Problem 15 - Algebra 2

Question

Write a recursive formula for each sequence. Then find the next term. $$ 6,1,-4,-9, \dots $$

Step-by-Step Solution

Answer

-14

Step 1: Identifying the pattern

By comparing each successive term, we can identify the pattern in the given sequence. For this sequence, it can be seen that we subtract 5 to get the next term from the current one i.e., 1 - 6 = -5, -4 - 1 = -5, -9 - (-4) = -5 and so on.

Step 2: Formulating the recursive formula

Given the pattern, we could write our recursive formula. A recursive formula allows us to find the nth term by relating it to its previous terms. For this type of sequence, the formula would typically be \( a_n = a_{n-1} - 5 \), which says that the next term can be found by subtracting 5 from the previous term.

Step 3: Finding the next term

Now, applying our recursive formula \( a_n = a_{n-1} - 5 \) where \( a_{n-1} \) is the last known term of -9, we can find the next term. When we calculate this, -9 - 5 equals -14. Therefore, the next term of the sequence is -14.

Key Concepts

Recursive Formula

In mathematics, a recursive formula is a way to define the elements of a sequence with respect to its previous elements. It means you can generate any term of the sequence by applying a rule to the term that comes before it.
For the sequence given above, each subsequent term is derived by subtracting a fixed number from its predecessor.
The initial part starts with identifying a pattern that repeats throughout the sequence. Once this pattern is recognized, it can be translated into a rule.
A typical recursive formula is expressed as:

• \( a_n = a_{n-1} + d \)

Where:

• \( a_n \) is the nth term
• \( a_{n-1} \) is the term before the nth
• \( d \) is the common difference between every two consecutive terms.

In our exercise, the difference \(-5\) was identified as the pattern. Thus, the recursive formula for this sequence is \( a_n = a_{n-1} - 5 \).
This is because, in this particular sequence, each term is obtained by subtracting 5 from the previous term. Whether you start with 6, 1, -4, or any other term, applying the formula ensures continuity.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two successive terms is constant. This constant is termed the 'common difference' and can be positive, negative, or zero.
In our example, the sequence \( 6, 1, -4, -9, \dots \) follows an arithmetic pattern. Each term is derived by subtracting 5 from the previous one, making the common difference \(-5\).
Features of an arithmetic sequence include:

• The sequence changes at a constant rate.
• It can increase, decrease, or be constant depending on the sign of the difference.
• Arithmetic sequences are straightforward in terms of recursive generation because each term is a fixed modification of the previous term.

Understanding arithmetic sequences allows you to predict any term in the sequence, provided you know at least one previous term and the common difference.

Sequence Pattern

Recognizing a sequence pattern is a foundational skill in understanding various mathematical sequences. A pattern indicates how terms of a sequence relate to each other. In an arithmetic sequence like ours, spotting the pattern involves identifying the constant difference between terms.
In the example sequence \(6, 1, -4, -9, \dots\), the key was to notice that each term decreases by 5. This repetitive change forms the sequence's pattern. Identifying this pattern allows not only for the construction of a recursive formula but also for predictions about future terms.
By understanding the pattern, students can:

• Formulate recursive and explicit formulas
• Predict missing terms in a sequence
• Understand more complex sequences and how they behave

Spotting patterns is not just useful in arithmetic sequences, but also in various other mathematical contexts, and is a crucial skill for problem-solving and analysis.