The sequence 4;9;x;37 is a quadratic sequence
1.1 calculate x
In a quadratic sequence, the 2nd differences are constant
terms of 1st sequence: 9-4, x-9, 37 - x
terms of 2nd sequence:
x-9 - 5, 37-x - (x-9)
or : x - 14, 46 - 2x
then x - 14 = 46 - 2x
3x = 60
x = 20
To calculate the value of "x" in the quadratic sequence 4, 9, x, 37, we need to find the pattern in the sequence and determine the underlying quadratic equation. Once we have the equation, we can substitute the given values to solve for "x".
Step 1: Find the difference between consecutive terms:
The difference between consecutive terms can provide information about the sequence's pattern.
9 - 4 = 5
x - 9 = d (where "d" is the common difference)
37 - x = d
The common difference, "d", is the same in both cases: 5.
Step 2: Write the quadratic equation:
In a quadratic sequence, the second term is related to the first term through a linear function, and the third term is related to the second term through a quadratic function.
Let's assume the quadratic equation for this sequence is:
an = a + (n - 1)d + c(n - 1)^2
Where:
an is the nth term
a is the first term
d is the common difference
c is the coefficient of the quadratic term
In our case:
a = 4
d = 5 (common difference)
Substituting these values into the equation, we get:
an = 4 + (n - 1)5 + c(n - 1)^2
Step 3: Use the given values to solve for "x":
We have:
x - 9 = d -> x = 9 + d = 9 + 5 = 14
37 - x = d -> x = 37 - d = 37 - 5 = 32
Now, substitute these values of "x" into the quadratic equation from Step 2:
When x = 14:
4 + (3 - 1)5 + c(3 - 1)^2 = 14
4 + 2(5) + 4c = 14
4 + 10 + 4c = 14
14 + 4c = 14
4c = 14 - 14
4c = 0
c = 0/4
c = 0
When x = 32:
4 + (4 - 1)5 + c(4 - 1)^2 = 32
4 + 3(5) + 9c = 32
4 + 15 + 9c = 32
19 + 9c = 32
9c = 32 - 19
9c = 13
c = 13/9
So, the quadratic equation for the given sequence is:
an = 4 + (n - 1)5 + (13/9)(n - 1)^2
To calculate x, substitute the value of n with the appropriate position of x in the sequence. Assuming that x is the 3rd term (n = 3):
x = 4 + (3 - 1)5 + (13/9)(3 - 1)^2
x = 4 + 2(5) + (13/9)(2)^2
x = 4 + 10 + (13/9)(4)
x = 14 + (13/9)(4)
x = 14 + (52/9)
x = 126/9 + 52/9
x = 178/9
Therefore, x is equal to 178/9 or approximately 19.778.