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, 2, 4, 8, 16, ......}

Verified step by step guidance

1. Observe the given sequence: {1, 2, 4, 8, 16, ...}. Notice the pattern in how each term relates to the previous term.
2. Identify the relationship between consecutive terms. For example, check how to get from 1 to 2, from 2 to 4, and so on. This will help you find the recurrence relation.
3. Express the recurrence relation in the form \(a_{n} = f(a_{n-1})\), where \(a_{n}\) is the current term and \(a_{n-1}\) is the previous term. Based on the pattern, determine the function \(f\).
4. Specify the initial index value, usually \(n=1\), and state the first term of the sequence \(a_1\) explicitly.
5. Write the complete recurrence relation including the initial condition, for example: \(a_1 = 1\) and \(a_n = 2 \times a_{n-1}\) 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, typically denoted as aₙ, where n indicates the term's position. Understanding how terms progress helps identify patterns or formulas.

Recurrence Relations

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). For example, aₙ = 2aₙ₋₁ defines each term as twice the previous term.

Initial Conditions

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