27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).


{1, 3, 9, 27, 81, ......}

Verified step by step guidance

1. Identify the pattern in the given sequence: {1, 3, 9, 27, 81, ...}. Notice how each term relates to the previous term.
2. Observe that each term is obtained by multiplying the previous term by 3. For example, 3 = 1 \times 3, 9 = 3 \times 3, and so on.
3. Express this relationship as a recurrence relation: \[a_{n} = 3 \times a_{n-1}\] where \[n \geq 2\].
4. Specify the initial condition, which is the first term of the sequence: \[a_1 = 1\].
5. Combine the recurrence relation and the initial condition to fully define the sequence: \[a_1 = 1, \quad a_n = 3a_{n-1} \text{ for } n \geq 2\].

Key Concepts

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific rule. Each number in the sequence is called a term, denoted as aₙ, where n indicates the term's position. Understanding how terms relate to their positions is essential for analyzing and generating sequences.

Recurrence Relation

A recurrence relation expresses each term of a sequence as a function of one or more previous terms. It provides a way to generate the sequence step-by-step, starting from initial term(s). Identifying the recurrence relation helps in understanding the pattern and behavior of the sequence.

Initial Conditions

Initial conditions specify the starting term(s) of a sequence, which are necessary to uniquely determine all subsequent terms using the recurrence relation. Without these values, the sequence cannot be fully generated or analyzed.