Problem 40 - Algebra 2

Question

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ -75,-68,-61,-54, \dots $$

Step-by-Step Solution

Answer

The next two terms in the sequence are -47 and -40. The explicit formula for the \(n\)th term is \(a_n = -75 + (n - 1) * 7\), and the recursive formula for the \(n\)th term is \(a_n = a_{n-1} + 7\), with \(a_1 = -75\). Both formulas are correctly identified as explicit and recursive, respectively.

Step 1: Determine the Common Difference

The common difference, \(d\), in an arithmetic sequence is the difference between any two successive terms. Here, the difference between -68 and -75 is 7. Similarly, the difference between -61 and -68, and -54 and -61, are both 7. So, \(d = 7\).

Step 2: Find the Next Two Terms

With the common difference identified as 7, the next term in the sequence can be found by adding 7 to the last known term, -54. Doing this yields -47 as the next term. Similarly, adding 7 to -47 yields -40 as the term after that. So, the next two terms are -47 and -40.

Step 3: Formulate the Explicit Formula

The explicit formula for an arithmetic sequence is \(a_n = a_1 + (n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference. Plugging in -75 for \(a_1\) and 7 for \(d\), we get \(a_n = -75 + (n - 1) * 7\).

Step 4: Formulate the Recursive Formula

The recursive formula for an arithmetic sequence is \(a_n = a_{n-1} + d\), where \(a_{n-1}\) is the previous term and \(d\) is the common difference. Plugging in 7 for \(d\), we get \(a_n = a_{n-1} + 7\), with \(a_1 = -75\).

Step 5: Classify the Formulas

The formula \(a_n = -75 + (n - 1) * 7\) is an explicit formula because it calculates the \(n\)th term directly, based on \(n\). The formula \(a_n = a_{n-1} + 7\), \(a_1 = -75\) is a recursive formula because it defines the \(n\)th term based on the previous term.

Key Concepts

Common Difference

In arithmetic sequences, the common difference is a crucial concept. It determines the amount you add to each term to get the next one. You can find it by subtracting any term from the succeeding one. For example, consider the sequence \(-75, -68, -61, -54, \dots\). Here, subtract -75 from -68 and find the **common difference** to be 7.

• This value, 7, remains constant throughout the sequence.
• Every time you move from one term to the next, you simply add this value.

The consistency of the common difference is what makes the sequence arithmetic. If you're ever unsure if a sequence is arithmetic, check if this difference is the same between all consecutive terms.

Explicit Formula

An explicit formula allows you to find the nth term of an arithmetic sequence directly. You do not need to know the previous term, making calculations quick and straightforward. The formula looks like this: \[ a_n = a_1 + (n - 1) \, d \] where:

• \(a_1\) is the first term of the sequence.
• \(n\) is the term position you want to find.
• \(d\) is the common difference.

Let's apply it to our sequence. With \(a_1 = -75\) and \(d = 7\), the formula becomes:\[ a_n = -75 + (n - 1) \, 7 \]You can use this formula to find any term in the sequence without prior terms. Just plug in the value of \(n\).

Recursive Formula

A recursive formula defines each term of an arithmetic sequence by referring to the previous term. It is like a guiding step-by-step instruction. For an arithmetic sequence, the recursive formula is:\[a_n = a_{n-1} + d\]where:

• \(a_{n-1}\) is the term before \(a_n\).
• \(d\) is the common difference.

For our sequence, we substitute \(d = 7\), so the formula becomes: \[a_n = a_{n-1} + 7\] with \(a_1 = -75\).This formula is handy when you want to build a sequence iteratively. Start from \(a_1\) and keep adding 7 to find the next terms.

Nth Term

The nth term of an arithmetic sequence represents any position in the sequence without needing to list all the previous terms. You can use either the explicit formula or the recursive formula to find it.

• If you need the nth term quickly, the explicit formula is your best bet as it only requires knowledge of the sequence's first term and common difference.
• If you are generating terms sequentially, the recursive formula is more practical as it builds from the preceding term.

By having both formulas at your fingertips, you can tackle tasks and problems involving the nth term of arithmetic sequences efficiently. Remember in our example, using the explicit formula: \[ a_n = -75 + (n - 1) \, 7 \]allows you to compute any term directly, saving steps and making calculations faster.