Quadratic sequence of 4, 16, 36, 64
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Answer:
To determine if the given sequence 4, 16, 36, 64 is a quadratic sequence, we need to check if there is a common difference between the terms.
Let's calculate the differences between consecutive terms:
16 - 4 = 12
36 - 16 = 20
64 - 36 = 28
The differences are not the same, so it does not follow a linear pattern.
Next, let's check if there is a common second difference. This will help determine if the sequence follows a quadratic pattern.
20 - 12 = 8
28 - 20 = 8
The second differences are the same, which suggests that the sequence follows a quadratic pattern.
To find the quadratic equation that represents this sequence, let's assume the formula is:
an = a * n^2 + b
Substituting the given terms into the equation:
For n = 1, a(1)^2 + b = 4
a + b = 4
For n = 2, a(2)^2 + b = 16
4a + b = 16
Solving the system of equations:
a + b = 4 (Equation 1)
4a + b = 16 (Equation 2)
From Equation 1, we get b = 4 - a
Substituting b in Equation 2:
4a + (4 - a) = 16
3a = 12
a = 4
Substituting a = 4 into Equation 1:
4 + b = 4
b = 0
Thus, the quadratic equation that represents the given sequence is:
an = 4n^2
Therefore, the next term in the sequence would be:
a5 = 4(5)^2 = 4(25) = 100