Problem 31 - Algebra 2

Question

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence. $$ a_{n}=a_{n-1}-17, \text { where } a_{1}=340 $$

Step-by-Step Solution

Answer

The formula is recursive. The first five terms of the sequence are 340, 323, 306, 289, and 272.

Step 1: Identifying the Type of Formula

Looking at the formula \(a_{n}=a_{n-1}-17\), we can see it refers to the term before it, which makes it recursive.

Step 2: Calculating the First Term

We don't need to calculate the first term, because the exercise already provides it. So, \(a_{1}=340\).

Step 3: Calculating the Second Term

Now we substitute \(n=2\) in the formula: \(a_{2}=a_{2-1}-17=a_{1}-17=340-17=323\).

Step 4: Calculating the Third Term

Again, substitute \(n=3\) in the formula: \(a_{3}=a_{3-1}-17=a_{2}-17=323-17=306\).

Step 5: Calculating the Fourth Term

Keep substituting, now \(n=4\): \(a_{4}=a_{4-1}-17=a_{3}-17=306-17=289\).

Step 6: Calculating the Fifth Term

And at last, substitute \(n=5\): \(a_{5}=a_{5-1}-17=a_{4}-17=289-17=272\).

Key Concepts

Sequence Formulas

In the world of mathematics, sequences play a vital role. Understanding them is crucial, and this begins with sequence formulas. Two main types of sequence formulas exist: explicit and recursive.

• **Explicit Formulas** provide a direct way to find any term in a sequence without referring to other terms. For example, if you know the starting point and the pattern, you can easily calculate any term by plugging numbers into a formula.
• **Recursive Formulas,** on the other hand, determine each term based on the previous ones. In these, you need the initial terms to start the process. Much like a chain reaction, each term generates the next.

In our original exercise, the formula is recursive: \(a_{n}=a_{n-1}-17\). Here, the formula tells us to subtract 17 from the previous term to get the next one. Recursive formulas often make sequences more intriguing, as each term builds on the last.

Arithmetic Sequences

An arithmetic sequence is a type of number sequence wherein each term after the first is found by adding a constant difference, known as the common difference, to the preceding term. These are linear by nature and often very easy to understand and recognize.
To determine if a sequence is arithmetic, look for that consistent difference between consecutive terms. Dive into the example given: the sequence determined by \(a_{n}=a_{n-1}-17\) clearly shows a common difference of -17.
This means for each term after the first, simply subtract 17 from the previous term.

• This approach simplifies calculations and allows predictions of any future terms without directly calculating preceding ones every time.
• Additionally, understanding that this is an arithmetic sequence assures you that the graph of the sequence describes a straight line with a constant downward slope due to the negative difference.

Sequence Terms

Each individual number in a sequence is known as a sequence term. Understanding and identifying these terms is critical to mastering sequences.
In a recursive sequence like ours, terms are generated step by step, using the rule provided. Starting from our initial term, \(a_{1}=340\):

• The second term, \(a_{2}\), is calculated as 323 since \(a_{2}=340-17\).
• The third term, \(a_{3}\), becomes 306, as \(a_{3}=323-17\).
• For the fourth term, \(a_{4}=306-17=289\), and finally,
• The fifth term is 272, calculated as \(a_{5}=289-17\).

In most cases, the first few terms are critical for setting the stage for the entire sequence. They provide insights into the sequence's behavior and are crucial for any pattern recognition or further sequence analysis. Knowing how to compute these terms confidently will aid extensively in more complex sequence explorations later on.