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 sequences do not converge, we need to examine the behavior of the terms in each sequence as n approaches infinity.
A sequence converges if and only if its terms approach a fixed value as n gets larger. Thus, if any of the options do not have terms approaching a fixed value, they do not converge.
Let's analyze each option:
A) P0 = 40, Pn+1 − Pn = 2.8 Pn (1− Pn/300) (n = 0,1,2, . . .)
To check if this sequence converges, we should examine how the terms behave as n approaches infinity. Unfortunately, we cannot determine this without additional information. We need to know the value of P0 and if the sequence is bounded.
B) P0 = 100, Pn+1 − Pn = 0.7Pn (1− Pn/480) (n = 0,1,2, . . .)
As with option A, we cannot determine the convergence of this sequence without additional information. We need to know the value of P0 and if the sequence is bounded.
C) P0 = 250, Pn+1 − Pn = 2.4Pn (1− Pn/420) (n = 0,1,2, . . .)
Similar to the previous options, we need more information to determine if this sequence converges or not.
D) an = (5−3n)/(7n + 12) (n = 0,1,2, . . .)
To check for convergence, we can examine the behavior of the expression as n approaches infinity. The denominator, 7n + 12, grows without bound as n increases, while the numerator, 5 - 3n, approaches negative infinity. Therefore, the terms of this sequence do not approach a fixed value, and it does not converge.
E) an = 50 /(5(0.2))^n (n = 0,1,2, . . .)
In this sequence, the denominator (5(0.2))^n grows exponentially as n increases. This means that the terms decrease exponentially as n increases. Since the terms are not approaching a fixed value, this sequence does not converge.
F) an = (8n^4 + 10n^2) / (4−3n^5) (n = 0,1,2, . . .)
We can examine the behavior of the terms by looking at the highest degree terms. The numerator, 8n^4, grows faster than the denominator, 4−3n^5, as n approaches infinity. Therefore, the terms of this sequence do not approach a fixed value, and it does not converge.
Based on this analysis, the three options that define sequences that do not converge are D, E, and F.