Problem 18 - Algebra and Trigonometry

Question

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=5 \text { and } a_{n}=3 a_{n-1}-1 \text { for } n \geq 2 $$

Step-by-Step Solution

Answer

The first four terms of the sequence are \( a_{1}=5 \), \( a_{2}=14 \), \( a_{3}=41 \), and \( a_{4}=122 \).

Step 1: Understand the Recursion formula

A recursion formula is a way of defining a sequence where each term \( a_{n} \) is defined as a function of its preceding term \( a_{n-1} \). Here the given recursive formula is \( a_{1}=5 \) and \( a_{n}=3 a_{n-1}-1 \) for \( n \geq 2 \). So starting from the second term, each term will be three times the preceding term minus one.

Step 2: Find the second term \( a_{2} \)

Now, to find the second term \( a_{2} \), substitute \( n=2 \) in the recursion formula \( a_{n}=3 a_{n-1}-1 \). Thus, we get \( a_{2}=3 a_{2-1}-1=3 a_{1}-1=3(5)-1=14 \). So the second term \( a_{2} \) is 14.

Step 3: Find the third term \( a_{3} \)

Similarly, to find the third term \( a_{3} \), substitute \( n=3 \) in the recursion formula \( a_{n}=3 a_{n-1}-1 \). Thus, we get \( a_{3}=3 a_{3-1}-1=3 a_{2}-1=3(14)-1=41 \). So the third term \( a_{3} \) is 41.

Step 4: Find the fourth term \( a_{4} \)

Lastly, to find the fourth term \( a_{4} \), substitute \( n=4 \) in the recursion formula \( a_{n}=3 a_{n-1}-1 \). Thus, we get \( a_{4}=3 a_{4-1}-1=3 a_{3}-1=3(41)-1=122 \). So the fourth term \( a_{4} \) is 122.

Key Concepts

Sequence

Understanding the concept of a sequence is fundamental when learning about recursive formulas. A sequence is essentially an ordered list of numbers. Each number in the list is referred to as a "term." Sequences can be finite or infinite.

One key aspect is that each term in a sequence follows a specific pattern or rule. In class, you might have encountered sequences like the arithmetic progression, where you add the same value to each term to get the next, or geometric sequences, where you multiply each term by the same factor. These straightforward patterns are different from recursive sequences—which we will discuss next—but the concept of following a rule to determine further terms remains common in all sequences.

Understanding whether a sequence has a hidden pattern or design is crucial. This understanding makes it possible to predict future terms without listing them all.

Recursive Formula

Recursive formulas provide a dynamic way to define sequences. Unlike other formulas, a recursive formula specifies how each term directly depends on the previous one. This is what makes it "recursive," as the formula refers back to itself.

In our exercise, we deal with a recursive sequence defined by two pieces – an initial term, and the rule for finding each subsequent term. The initial term given is \( a_1 = 5 \). The recursive formula provided is \( a_n = 3a_{n-1} - 1 \) for \( n \geq 2 \).

• The \( a_1 \) tells us the first number in the sequence.
• The \( a_n \) formula helps in finding further terms as long as you know the previous term.

The beauty of the recursive formula is its ability to compactly express potentially complex sequences. However, it requires a bit of calculation for each term, which is both a learning point and a practice for understanding dependencies in sequences.

Sequence Terms

Finding sequence terms using a recursive formula involves substituting values step-by-step. This is a methodical process—it requires using earlier terms to calculate subsequent ones, adhering strictly to the given pattern.

Let's unpack our previous example a bit further:

• Start with \( a_1 = 5 \).
• Using the recursive formula \( a_n = 3a_{n-1} - 1 \), find the second term: \( a_2 = 3a_1 - 1 = 14 \).
• Next, for the third term, \( a_3 = 3a_2 - 1 = 41 \).
• Finally, for the fourth term, \( a_4 = 3a_3 - 1 = 122 \).

This approach underlines an essential part of recursion—consistently leveraging prior results to determine subsequent results. While it seems straightforward, recognizing each calculated term takes time and practice. Building confidence with recursion changes how you perceive mathematical patterns, shifting your focus from direct formulas to indirect computation through recursion.