Suppose the sequence {aₙ}⁽∞⁾ₙ₌₀ is defined by the recurrence relation
aₙ₊₁ = ⅓aₙ + 6;a₀ = 3.


b.Explain why {aₙ}⁽∞⁾ₙ₌₀ converges and find the limit.

Verified step by step guidance

1. Recognize that the sequence \( \{a_n\} \) is defined by a linear recurrence relation of the form \( a_{n+1} = r a_n + c \), where \( r = \frac{1}{3} \) and \( c = 6 \). Since \( |r| = \frac{1}{3} < 1 \), the sequence is a contraction and will converge to a fixed point.
2. To find the limit \( L \) of the sequence, assume it exists and satisfies the recurrence relation in the limit, so \( L = \frac{1}{3} L + 6 \). This is because as \( n \to \infty \), \( a_n \to L \) and \( a_{n+1} \to L \).
3. Solve the equation for \( L \): \( L = \frac{1}{3} L + 6 \). Rearranging gives \( L - \frac{1}{3} L = 6 \), which simplifies to \( \frac{2}{3} L = 6 \).
4. Multiply both sides by \( \frac{3}{2} \) to isolate \( L \), yielding \( L = 6 \times \frac{3}{2} \).
5. Interpret the result: since the sequence converges to \( L \), this value is the fixed point of the recurrence relation and represents the long-term behavior of \( \{a_n\} \).

Key Concepts

Recurrence Relations

A recurrence relation defines each term of a sequence based on previous terms. Understanding how to manipulate and solve these relations is essential to analyze the behavior of sequences, such as finding explicit formulas or limits.

Convergence of Sequences

A sequence converges if its terms approach a specific finite value as the index goes to infinity. Determining convergence involves analyzing the long-term behavior of the sequence, often by examining the recurrence relation or using limit properties.

Finding Limits of Linear Recurrence Sequences

For linear recurrence relations like aₙ₊₁ = r aₙ + c with |r| < 1, the sequence converges to the fixed point L = c / (1 - r). This limit is found by setting aₙ₊₁ = aₙ = L and solving the resulting equation.