Choose the THREE options that define sequences that do not converge.
Options
A P0 = 40, Pn+1 − Pn = 2.8 Pn (1− Pn/300) (n = 0,1,2, . . .)
B P0 = 100, Pn+1 − Pn = 0.7Pn (1− Pn/480) (n = 0,1,2, . . .)
C P0 = 250, Pn+1 − Pn = 2.4Pn (1− Pn/420) (n = 0,1,2, . . .)
D an = (5−3n)/(7n + 12) (n = 0,1,2, . . .)
E an =50 /(5(0.2))^n (n = 0,1,2, . . .)
F an =(8n^4 + 10n^2) / (4−3n^5) (n = 0,1,2, . . .)
To determine which options define sequences that do not converge, we need to analyze the behavior of each sequence.
A sequence converges if its terms approach a specific value as n increases. On the other hand, a sequence diverges if its terms do not approach a specific value or if they oscillate without settling down.
Let's analyze each option:
A) P₀ = 40, Pₙ₊₁ − Pₙ = 2.8Pₙ(1− Pₙ/300) (n = 0,1,2, . . .)
This recursive formula appears complicated, but we can still explore its behavior. You could start by calculating a few terms of the sequence to see if it approaches a specific value.
B) P₀ = 100, Pₙ₊₁ − Pₙ = 0.7Pₙ(1− Pₙ/480) (n = 0,1,2, . . .)
Similarly, calculate a few terms of the sequence to see if it approaches a specific value or diverges.
C) P₀ = 250, Pₙ₊₁ − Pₙ = 2.4Pₙ(1− Pₙ/420) (n = 0,1,2, . . .)
Analyze this sequence's behavior by calculating a few terms and determining if there is convergence or divergence.
D) aₙ = (5−3n)/(7n + 12) (n = 0,1,2, . . .)
Consider the behavior of this sequence by calculating some terms. If it approaches a specific value, it converges; otherwise, it diverges.
E) aₙ = 50 /(5(0.2))^n (n = 0,1,2, . . .)
Analyze the terms of this sequence by calculating a few of them. If they approach a specific value, the sequence converges; if not, it diverges.
F) aₙ = (8n⁴ + 10n²) / (4−3n⁵) (n = 0,1,2, . . .)
Examine this sequence by calculating some terms. If they approach a specific value, the sequence converges; if not, it diverges.
After analyzing the behavior of each sequence, identify the three options that define sequences that do not converge.