For the sequence below what are the next 3 elements and why? Please help ASAP!!! :(
3,1,4,5,9,14, ___ , ___ , ___
Looks like every element is dependent on the previous two elements.
hint: read about the Fibonacci sequence
To find the next three elements in the given sequence, we need to observe the pattern or rule that generates the sequence. Let's analyze the sequence step-by-step.
The difference between consecutive terms can give us some insights. Let's calculate the difference between each pair of terms:
1st difference: 1 - 3 = -2
2nd difference: 4 - 1 = 3
3rd difference: 5 - 4 = 1
4th difference: 9 - 5 = 4
5th difference: 14 - 9 = 5
From the 2nd difference onwards, we can see that the differences are not constant. However, let's see if there is a pattern within the differences:
1st difference: -2 -> This seems to be decreasing by 1 each time.
2nd difference: 3 -> This seems to be constant.
3rd difference: 1 -> This is constant.
4th difference: 4 -> This seems to be increasing by 1 each time.
5th difference: 5 -> This seems to be increasing by 1 each time.
Based on these patterns, it appears that the sequence follows a quadratic pattern. Let's find the formula for the sequence based on this pattern.
Considering the pattern of the 2nd differences, we can use a quadratic equation in the form of:
An^2 + Bn + C
Using the given terms and their positions in the sequence, we can substitute the values into the equation.
When n = 1, the term is 3, which gives us the equation:
A(1)^2 + B(1) + C = 3 => A + B + C = 3 (Equation 1)
When n = 2, the term is 1, which gives us the equation:
A(2)^2 + B(2) + C = 1 => 4A + 2B + C = 1 (Equation 2)
When n = 3, the term is 4, which gives us the equation:
A(3)^2 + B(3) + C = 4 => 9A + 3B + C = 4 (Equation 3)
We have three equations with three variables (A, B, C). By solving these equations simultaneously, we can find the values of A, B, and C.
Solving Equations 1, 2, and 3, we find:
A = -1
B = 6
C = -2
Now that we have found A, B, and C, we can use the equation An^2 + Bn + C to find the next three elements in the sequence.
When n = 7:
(-1)(7)^2 + (6)(7) + (-2) = -49 + 42 - 2 = -9
When n = 8:
(-1)(8)^2 + (6)(8) + (-2) = -64 + 48 - 2 = -18
When n = 9:
(-1)(9)^2 + (6)(9) + (-2) = -81 + 54 - 2 = -29
Therefore, the next three elements in the sequence are:
-9, -18, -29