Use iteration to guess an explicit formula for the sequence.
Pk=Pk-1+2*3^k
P1=2
P_k - P_(k-1) = 2*3^k
Sum both sides from k = 2 to k = n. Left hand side becomes:
P_2 - P_1 + P_3 - P_2 + P_4 - P_3 + P_5 - P_4 + .....+ P_n - P_(n-1) =
P_n - P_1
Right hand side is:
2* [3^2-3^(n+1)]/(1-3) =3^(n+1) - 9
P_n - P_1 = 3^(n+1) - 9 --->
P_n = 3^(n+1) -7
To guess an explicit formula for the sequence, let's start by writing out the first few terms of the sequence:
P1 = 2
P2 = P1 + 2 * 3^2 = 2 + 2 * 9 = 20
P3 = P2 + 2 * 3^3 = 20 + 2 * 27 = 74
P4 = P3 + 2 * 3^4 = 74 + 2 * 81 = 236
Observing the sequence, we can see that each term Pk is obtained by adding 2 times the value of 3 raised to the power of k. This suggests that the explicit formula for the sequence is:
Pk = 2 + 2 * 3^k
To guess an explicit formula for the sequence, we can iterate through a few terms of the sequence until we find a pattern. Let's start by calculating the first few terms of the sequence using the given recurrence relation:
P1 = 2
Using the recurrence relation:
P2 = P1 + 2 * 3^2 = 2 + 2 * 9 = 20
P3 = P2 + 2 * 3^3 = 20 + 2 * 27 = 74
P4 = P3 + 2 * 3^4 = 74 + 2 * 81 = 236
From these initial terms, we can observe the pattern. It looks like the terms in the sequence are increasing rapidly, with each term depending on the previous term multiplied by a certain power of 3, and then adding a constant.
To guess an explicit formula for this sequence, let's break down the recurrence relation:
P(k) = P(k-1) + 2 * 3^k
Breaking it down further:
P(k) = P(k-1) + 2 * (3^k) = P(k-1) + 2 * (3 * 3^(k-1))
Now, we can see that this can be simplified as:
P(k) = P(k-1) + 2 * 3 * (3^(k-1))
This indicates that each term in the sequence is the sum of the previous term and a constant value, which is 2 * 3 * (3^(k-1)).
Now, we need to find the explicit formula for the sequence:
P1 = 2
From the pattern we observed, it seems like the explicit formula for the sequence is given by:
P(k) = P(1) + 2 * 3 * (3^(k-1))
Replacing P(1) with its value, the explicit formula becomes:
P(k) = 2 + 2 * 3 * (3^(k-1))
So, the explicit formula for the sequence is P(k) = 2 + 2 * 3 * (3^(k-1)).