Need help I have to find a pattern for
8, -1, 2, -1
I know the answer is 1/3 but have no idea how I get there...I am trying to multiply each by -1/2 but I can't get the 1/3 at the end for some reason
and I have the similar problem as well:
8, -4, 2, -1
To find the pattern for the sequence 8, -1, 2, -1, we need to look for the relationship between consecutive terms.
In this case, to get from 8 to -1, we can see that we subtracted 9 (-1 - 8 = -9). Then, to get from -1 to 2, we can see that we added 3 (2 - (-1) = 3). Finally, to get from 2 to -1, we subtracted 3 again (-1 - 2 = -3).
From these observations, we can see that the pattern is changing the value by alternately adding or subtracting 9 and 3.
Therefore, to find the next term in the sequence, we subtract 9 from -1. (-1 - 9 = -10). And to find the term after that, we add 3 to -10 (-10 + 3 = -7).
The extended sequence then becomes: 8, -1, 2, -1, -10, -7, ...
Now, let's examine how to obtain the answer of 1/3.
Starting with the sequence: 8, -1, 2, -1, -10, -7, ...
To find the pattern, let's subtract each term from the previous term:
-1 - 8 = -9
2 - (-1) = 3
-1 - 2 = -3
-10 - (-1) = -9
-7 - (-10) = 3
Notice a pattern? The differences between consecutive terms are alternating between -9 and 3.
Now, let's examine the differences between the differences (second differences):
3 - (-9) = 12
-3 - 3 = -6
-9 - (-3) = -6
The second differences are constant (-6), which suggests that the original sequence is a quadratic sequence.
Using this information, we can set up a quadratic expression to find the formula for the sequence. Let n represent the index of the term:
a(n) = an^2 + bn + c
To find the values of a, b, and c, let's plug in three terms from the sequence:
a(1) = 8
a(2) = -1
a(3) = 2
Using these values, we can set up three equations:
8 = a(1)^2 + b(1) + c
-1 = a(2)^2 + b(2) + c
2 = a(3)^2 + b(3) + c
Simplifying these equations, we get:
8 = a + b + c
-1 = 4a + 2b + c
2 = 9a + 3b + c
To solve these equations, we can use various methods such as substitution or elimination. Solving this system of equations will give us the values of a, b, and c, which will allow us to find the general formula for the sequence.
I hope this explanation helps you understand the process of finding patterns in sequences. If you have any further questions, feel free to ask!