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, . . .)
Thanks.
To determine which options define sequences that do not converge, we need to analyze each option and see if there are any patterns that suggest divergence.
Option A: P0 = 40, Pn+1 − Pn = 2.8 Pn (1− Pn/300) (n = 0,1,2, . . .)
This sequence has a recursive formula, which means we need to calculate each term using the previous term. To determine convergence, we would need to analyze the behavior of the terms in the long run. Since this cannot be deduced without explicitly calculating the terms, we cannot conclude whether this sequence converges or diverges.
Option B: P0 = 100, Pn+1 − Pn = 0.7Pn (1− Pn/480) (n = 0,1,2, . . .)
Similar to Option A, this sequence has a recursive formula, and we cannot determine convergence without explicitly calculating the terms.
Option C: P0 = 250, Pn+1 − Pn = 2.4Pn (1− Pn/420) (n = 0,1,2, . . .)
Again, this sequence has a recursive formula, and we cannot conclude convergence without explicitly calculating the terms.
Option D: an = (5−3n)/(7n + 12) (n = 0,1,2, . . .)
To analyze this sequence, we can observe its behavior as n approaches infinity. In this case, the numerator approaches a finite value (-3), while the denominator grows indefinitely. Therefore, this sequence does not converge.
Option E: an = 50 /(5(0.2))^n (n = 0,1,2, . . .)
Similar to Option D, we can observe the behavior as n approaches infinity. In this case, the denominator grows exponentially with an exponent greater than 1. Therefore, as n grows, the terms of this sequence become infinitesimally small, indicating divergence.
Option F: an =(8n^4 + 10n^2) / (4−3n^5) (n = 0,1,2, . . .)
By observing the behavior as n approaches infinity, we can see that the numerator grows faster than the denominator, leading to a sequence that does not converge.
From the analysis above, the three options that define sequences that do not converge are D, E, and F.