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 check if the sequences are bounded or if they oscillate.
1. Option A: P0 = 40, Pn+1 − Pn = 2.8 Pn (1− Pn/300)
We can observe that the expression 2.8 Pn (1− Pn/300) is always positive for Pn > 0. Therefore, the terms of the sequence will keep increasing indefinitely. This sequence is unbounded and does not converge.
2. Option B: P0 = 100, Pn+1 − Pn = 0.7Pn (1− Pn/480)
Similar to Option A, the expression 0.7Pn (1− Pn/480) is always positive for Pn > 0. Thus, the terms of the sequence will also increase indefinitely. This sequence is unbounded and does not converge.
3. Option C: P0 = 250, Pn+1 − Pn = 2.4Pn (1− Pn/420)
Again, the expression 2.4Pn (1− Pn/420) is always positive for Pn > 0, indicating that the terms of the sequence will continue to increase indefinitely. Thus, this sequence is unbounded and does not converge.
4. Option D: an = (5−3n)/(7n + 12)
To determine if this sequence converges, we can take the limit of the sequence as n approaches infinity. By evaluating the limit, we find that it approaches zero. Therefore, this sequence converges.
5. Option E: an = 50 /(5(0.2))^n
Simplifying the expression, we have an = 50 / (1^n). Since any number raised to the power of zero is equal to 1, the terms of this sequence are constant at 50. Therefore, this sequence converges.
6. Option F: an = (8n^4 + 10n^2) / (4−3n^5)
By taking the limit of this sequence as n approaches infinity, we find that it approaches infinity. Thus, this sequence is unbounded and does not converge.
Based on the analysis, the three options that define sequences that do not converge are:
- Option A: P0 = 40, Pn+1 − Pn = 2.8 Pn (1− Pn/300)
- Option B: P0 = 100, Pn+1 − Pn = 0.7Pn (1− Pn/480)
- Option C: P0 = 250, Pn+1 − Pn = 2.4Pn (1− Pn/420)