21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 2aₙ; a₁ = 2

Verified step by step guidance

1. Identify the given recurrence relation: \(a_{n+1} = 2a_n\) with the initial term \(a_1 = 2\).
2. Use the initial term to find the second term by substituting \(n=1\) into the recurrence relation: \(a_2 = 2a_1\).
3. Find the third term by substituting \(n=2\): \(a_3 = 2a_2\).
4. Find the fourth term by substituting \(n=3\): \(a_4 = 2a_3\).
5. Write out the first four terms as \(a_1\), \(a_2\), \(a_3\), and \(a_4\) using the values found in the previous steps.

Key Concepts

Recurrence Relations

A recurrence relation defines each term of a sequence using one or more previous terms. It provides a way to generate the sequence step-by-step, starting from given initial conditions.

Initial Conditions

Initial conditions specify the starting values of a sequence, which are necessary to compute subsequent terms using the recurrence relation. Without these, the sequence cannot be uniquely determined.

Sequence Generation

Sequence generation involves applying the recurrence relation repeatedly to find terms beyond the initial ones. For example, using aₙ₊₁ = 2aₙ and a₁ = 2, we calculate a₂, a₃, and a₄ by substitution.