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 the sequences that do not converge, we need to analyze each option and understand the behavior of the sequences.
A sequence converges if its terms approach a finite limit as n approaches infinity. On the other hand, a sequence diverges if its terms do not approach a limit or approach infinity or negative infinity.
Option A:
P0 = 40
Pn+1 - Pn = 2.8Pn(1 - Pn/300)
This is a recursive formula. To check if it converges, we need to analyze the behavior of the terms as n increases. Unfortunately, it is not easy to determine the limit analytically. To simplify the problem, we can graph the sequence and observe its behavior. If the graph shows no clear trend towards a specific value, it is likely that the sequence does not converge.
Option B:
P0 = 100
Pn+1 - Pn = 0.7Pn(1 - Pn/480)
Similar to option A, this is a recursive formula that requires a graphical analysis to determine convergence.
Option C:
P0 = 250
Pn+1 - Pn = 2.4Pn(1 - Pn/420)
Again, a recursive formula that requires a graphical analysis to determine convergence.
Option D:
an = (5-3n)/(7n + 12)
To determine if this sequence converges, we can examine its behavior as n approaches infinity. We can calculate the limit of the terms as n goes to infinity. If the limit exists and is finite, the sequence converges. If not, it diverges.
Option E:
an = 50/(5(0.2))^n
Simplifying, we have an = 50/((1)^n). The denominator is always constant, so the terms of the sequence do not approach any specific value as n increases. Thus, the sequence diverges.
Option F:
an = (8n^4 + 10n^2)/(4-3n^5)
Similar to option D, we can check the limit of the terms as n goes to infinity. If the limit exists and is finite, the sequence converges. If not, it diverges.
By analyzing options A, B, C, D, E, and F, we can determine which ones do not converge based on the behavior of their terms.