Question

84–87. {Use of Tech} Sequences by recurrence relations
The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.

a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.
b.Use analytical methods to find the limit of the sequence.

aₙ₊₁ = 2aₙ(1 − aₙ);a₀ = 0.3

Accepted Answer

Step 1: Understand the recurrence relation given: $$a_{n+1} = 2a_n(1 - a_n)$$ with initial term $$a_0 = 0.3$. This defines each term based on the previous term. Step 2: Calculate the first three terms explicitly to observe the behavior of the sequence: compute a_1$ = 2a_0(1 - a_0$)$$, then $$a_2 = 2a_1(1$$ - $$a_1$$)$$, and a_3$ = 2a_2(1 - a_2$). $$Compare these values to determine if the sequence is nondecreasing (each term greater than or equal to the previous) or nonincreasing (each term less than or equal to the previous). Step 3: To find the limit analytically, assume the sequence converges to a limit $$L. $$Since the sequence converges, the limit satisfies the same recurrence relation in the steady state: $$L = 2L(1 - L). $$Step 4: Solve the equation $$L = 2L(1 - L)$$ for $$L. $$This will involve rearranging the equation into a quadratic form and finding the roots. Step 5: Analyze the roots found in Step 4 and use the initial condition and monotonicity from Step 2 to determine which root is the actual limit of the sequence.

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Key Concepts

Recurrence Relations

Monotonicity and Boundedness

Finding Limits of Sequences Analytically