91Ó°ÊÓ

Understanding the sequence pattern is the first step to solving problems involving sequences. In this exercise, we are given a sequence: \( \frac{1}{3}, 1, 3, 9, \dots\ \).
We need to identify how each term is related to the previous one. Upon close observation, you can see the second term (1) is obtained by multiplying the first term \( \frac{1}{3} \) by 3. Similarly, the third term (3) is the second term multiplied by 3, and the pattern continues.
This pattern tells us that each term in the sequence is 3 times the term immediately before it.
Recognizing such patterns requires practice. Here are a few tips to help you identify sequence patterns:

• Look for common differences or ratios between terms.
• Check if terms are increasing or decreasing by a common factor.
• Write down the sequence and manually calculate differences or ratios for a few initial terms.

Spotting these patterns lays the groundwork for establishing a recursive relationship.

A recursive relationship expresses each term of the sequence as a function of its preceding term(s). Once we identify the sequence pattern, we can write a recursion formula.
In our sequence \( \frac{1}{3}, 1, 3, 9, \dots \), we observed that each term is 3 times the previous term. Hence, the recursive relationship can be expressed as:
\[ a_{n+1} = 3a_n \]
where \( a_{n+1} \) denotes the \( (n+1)^{th} \) term, and \( a_n \) is the n-th term.
To generalize, a recursive formula will often require:

• A clear pattern or ratio identified between terms.
• A functional relationship between consecutive terms.

This recursive relationship helps us compute any term in the sequence, provided we know the previous term.

Defining the initial term is crucial for making use of the recursion formula. The initial term is the first element in the sequence, which forms the base case for our recursion.
In our example sequence \( \frac{1}{3}, 1, 3, 9, \dots \), the initial term \( a_1 \) is clearly \( \frac{1}{3} \).
With the initial term set, we can now apply our recursive relationship to generate subsequent terms. The complete recursive formula becomes:
\[ a_{n+1} = 3a_n \]
with the initial term \( a_1 = \frac{1}{3} \).
In summary, to fully define and work with a recursion formula, always keep these points in mind:

• Identify the initial term clearly.
• Combine it with the recursive relationship discovered.

This approach ensures you can compute any term in the sequence starting from the initial term and using the recursive formula repeatedly.