Which recursive sequence would produce the sequence 6, -26, 102, ...6,β26,102,...?
more like
a_1 = 6
a_n+1 = -4a_n - 2
To determine the recursive sequence that produces the given sequence {6, -26, 102, ...}, we need to analyze the pattern between the terms.
1. Notice that the first term is 6.
2. The second term, -26, is obtained by multiplying the previous term by -4 and subtracting 10: -4 * 6 - 10 = -24 -10 = -34.
3. The third term, 102, is obtained by multiplying the previous term by -4 and adding 10: -4 * -26 + 10 = 104 + 10 = 114.
Therefore, the recursive sequence that generates the given sequence is:
a1 = 6
an = -4 * an-1 + 10, where n > 1, and an represents the nth term of the sequence.
To find the recursive sequence that produces the given sequence 6, -26, 102, ..., we need to examine the pattern and identify the relationship between consecutive terms.
Let's start by looking at the differences between consecutive terms:
- (-26 - 6) = -32
- (102 - (-26)) = 128
We can observe that the differences are increasing by a factor of 4, which suggests that our recursive relationship involves multiplying each term by 4.
Now, let's try multiplying each term by 4:
- 6 * 4 = 24
- -26 * 4 = -104
- 102 * 4 = 408
The resulting sequence is 24, -104, 408, ..., which is very close to the original sequence.
To refine it further, we need to consider another factor that may affect the sequence. Let's look at the differences between the consecutive terms of our new sequence:
- (-104 - 24) = -128
- (408 - (-104)) = 512
Once again, we observe that the differences are increasing, but this time by a factor of 8. Therefore, the recursive relationship for our sequence can be determined as follows:
- Start with 6 as the first term.
- Each subsequent term can be found by multiplying the previous term by 4 and then subtracting 32.
This recursive relationship can be summarized as:
ππ = 4ππβ1 β 32, where π0 = 6.
Using this formula, you can generate the given sequence: 6, -26, 102, ... as well as the subsequent terms in the sequence.