Find the first 6 terms of the sequence b^1 = 5. B^n = 2b(n -1) when n>1. Then write an explicit definition of the sequence and use it to find the 12th term.
Bn = 5*2^(n-1)
B12 = 5*2^11 = 10240
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To find the first 6 terms of the sequence, we can use the recursive formula provided and work our way up from the initial term.
Given:
b^1 = 5
For n > 1:
B^n = 2B(n-1)
First term (n=1): b^1 = 5
We can now calculate the remaining terms using the recursive formula:
Second term (n=2): B^2 = 2B(2-1) = 2B^1 = 2 * 5 = 10
Third term (n=3): B^3 = 2B(3-1) = 2B^2 = 2 * 10 = 20
Fourth term (n=4): B^4 = 2B(4-1) = 2B^3 = 2 * 20 = 40
Fifth term (n=5): B^5 = 2B(5-1) = 2B^4 = 2 * 40 = 80
Sixth term (n=6): B^6 = 2B(6-1) = 2B^5 = 2 * 80 = 160
So, the first 6 terms of the sequence are: 5, 10, 20, 40, 80, 160.
To write an explicit definition of the sequence, we need to find a formula that directly gives the nth term of the sequence without relying on previous terms. Let's analyze the pattern:
n | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|----|----|
Bn | 5 | 10 | 20 | 40 | 80 | 160 |
We can observe that each term is obtained by doubling the previous term. So, we can write the explicit definition of the sequence as follows:
Bn = 5 * 2^(n-1)
Now, let's use this explicit definition to find the 12th term:
B12 = 5 * 2^(12-1) = 5 * 2^11 = 5 * 2048 = 10240
Therefore, the 12th term of the sequence is 10240.