To find an arithmetic sequence function based on the given points \((0, 14)\), \((1, 20)\), and \((2, 27)\), we can start by determining the differences between the consecutive terms.
- The first term \( a_0 = 14 \).
- The second term \( a_1 = 20 \).
- The third term \( a_2 = 27 \).
Next, we can calculate the differences:
-
The difference between the second term and the first term is: \[ a_1 - a_0 = 20 - 14 = 6 \]
-
The difference between the third term and the second term is: \[ a_2 - a_1 = 27 - 20 = 7 \]
Since the differences are not constant (6 and 7), the sequence is not arithmetic. We will determine the explicit function that fits these points.
Let's denote \( f(x) \) as the function we want to find. Since the points do not form an arithmetic sequence, it’s more appropriate to use a quadratic function of the form: \[ f(x) = ax^2 + bx + c \]
Using the points, we can create a system of equations:
-
For \( x = 0 \): \[ f(0) = c = 14 \]
-
For \( x = 1 \): \[ f(1) = a(1)^2 + b(1) + c = 20 \implies a + b + 14 = 20 \implies a + b = 6 \]
-
For \( x = 2 \): \[ f(2) = a(2)^2 + b(2) + c = 27 \implies 4a + 2b + 14 = 27 \implies 4a + 2b = 13 \implies 2a + b = 6.5 \]
Now, we have the following system of equations:
- \( a + b = 6 \)
- \( 2a + b = 6.5 \)
We can solve this system. Let's subtract the first equation from the second: \[ (2a + b) - (a + b) = 6.5 - 6 \implies a = 0.5 \]
Now substitute \( a = 0.5 \) into the first equation: \[ 0.5 + b = 6 \implies b = 5.5 \]
Now we have:
- \( a = 0.5 \)
- \( b = 5.5 \)
- \( c = 14 \)
Thus, the function is: \[ f(x) = 0.5x^2 + 5.5x + 14 \]
This quadratic function fits the points \((0, 14)\), \((1, 20)\), and \((2, 27)\).
4 answers