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, . . .)
In order to determine which sequences do not converge, we need to examine each option and analyze the behavior of the sequence.
A sequence converges if its terms get arbitrarily close to a specific value as n (the term number) increases. If the terms do not approach a specific value or instead exhibit erratic behavior, the sequence does not converge.
Let's analyze each option:
Option A: P0 = 40, Pn+1 − Pn = 2.8 Pn (1− Pn/300) (n = 0,1,2, . . .)
Here, the recurrence relation involves the term Pn depending on the previous term Pn-1. To determine if this sequence converges, we need to check if it approaches a specific value over time. We can start by calculating the first few terms and observing their behavior. If the terms start oscillating or growing without bounds, the sequence does not converge.
Option B: P0 = 100, Pn+1 − Pn = 0.7Pn (1− Pn/480) (n = 0,1,2, . . .)
Similar to the previous option, we need to check the behavior of the sequence to determine if it converges or not. Calculate the first few terms and observe if they approach a specific value or if they exhibit erratic behavior.
Option C: P0 = 250, Pn+1 − Pn = 2.4Pn (1− Pn/420) (n = 0,1,2, . . .)
Again, examine the behavior of the terms in this sequence to see if they approach a specific value or display erratic behavior.
Option D: an = (5−3n)/(7n + 12) (n = 0,1,2, . . .)
To determine if this sequence converges, try calculating the first few terms and see if they converge to a specific value or not.
Option E: an =50 /(5(0.2))^n (n = 0,1,2, . . .)
Perform a similar analysis to the previous options by calculating the first few terms and observing their behavior.
Option F: an =(8n^4 + 10n^2) / (4−3n^5) (n = 0,1,2, . . .)
Apply the same process of calculating the first few terms and examining their behavior to determine if this sequence converges or not.
By analyzing the behavior of each sequence and determining if their terms approach a specific value, you can identify the three options that do not converge.