Calculate 20 terms for the sequence P0=-.5 and k=1.8, P0=.5 and k=2.8, and P0=.3 and k=1.8. Does the limit depend on the choice of P0? Does it depend on the choice of k?
To calculate the terms for the given sequence, we can start with the initial term P0 and then use the recursive formula to find the next terms.
The recursive formula for the sequence is:
Pn = k * Pn-1 * (1 - Pn-1)
where Pn represents the nth term of the sequence and Pn-1 represents the previous term.
Let's calculate the terms for each given sequence with their respective values of P0 and k.
For the first sequence (P0 = -0.5 and k = 1.8):
P1 = 1.8 * (-0.5) * (1 - (-0.5)) = 1.8 * (-0.5) * 1.5 = -1.35
P2 = 1.8 * (-1.35) * (1 - (-1.35)) = 1.8 * (-1.35) * 2.35 ≈ -4.89
Continuing this process, we can calculate the remaining terms for this sequence.
For the second sequence (P0 = 0.5 and k = 2.8):
P1 = 2.8 * 0.5 * (1 - 0.5) = 2.8 * 0.5 * 0.5 = 0.7
P2 = 2.8 * 0.7 * (1 - 0.7) = 2.8 * 0.7 * 0.3 ≈ 0.588
Continuing this process, we can calculate the remaining terms for this sequence as well.
For the third sequence (P0 = 0.3 and k = 1.8):
P1 = 1.8 * 0.3 * (1 - 0.3) = 1.8 * 0.3 * 0.7 ≈ 0.378
P2 = 1.8 * 0.378 * (1 - 0.378) = 1.8 * 0.378 * 0.622 ≈ 0.511
Continuing the calculation, we can find the rest of the terms for this sequence.
Now, let's analyze if the limit depends on the choice of P0 and k.
The limit of a sequence is the value it converges to as the terms progress towards infinity. In this case, we can calculate the limit by finding the value towards which the terms of the sequence are approaching.
To determine if the limit depends on the choice of P0 and k, we need to observe if the sequence converges to the same value for different choices of P0 and k.
By calculating a sufficient number of terms for each sequence, we can observe if the terms stabilize around a specific value. If the terms of each sequence converge to the same value regardless of the initial term P0 and the value of k, then the limit does not depend on the choice of P0 and k.
On the other hand, if the terms do not converge to the same value for different combinations of P0 and k, then the limit does depend on the choice of P0 and k.
By calculating the terms for the given combinations of P0 and k, analyze if the sequences converge to consistent values to determine whether the limit depends on P0 or k.