Question
Find the function of the arithmetic sequence with the folowing points
(0, 14), (1, 20), (2, 27)
(0, 14), (1, 20), (2, 27)
Answers
GPT-4o mini
An arithmetic sequence can be represented as a linear function of the form \( f(x) = mx + b \), where \( m \) is the common difference and \( b \) is the first term when \( x = 0 \).
To find the function based on the given points, we can first find the common difference \( m \):
1. From \( (0, 14) \) to \( (1, 20) \):
\[
m = 20 - 14 = 6
\]
2. From \( (1, 20) \) to \( (2, 27) \):
\[
m = 27 - 20 = 7
\]
It seems there’s an inconsistency here with the differences calculated. To check, let’s look at the points and determine if the arithmetic sequence is consistently applied:
- From \( f(0) = 14 \) to \( f(1) = 20 \) is indeed \( 6 \).
- From \( f(1) = 20 \) to \( f(2) = 27 \) is \( 7 \).
This inconsistency means that these points aren't forming a perfect arithmetic sequence with a uniform difference. It looks like there’s a misunderstanding of the arithmetic sequence nature.
To find a quadratic polynomial that fits these points, we will instead assume \( f(x) = ax^2 + bx + c \). Since we have three points, we can set up a system of equations using them:
1. From the point \( (0, 14) \):
\[
c = 14
\]
2. From the point \( (1, 20) \):
\[
a(1)^2 + b(1) + c = 20 \quad \Rightarrow \quad a + b + 14 = 20 \quad \Rightarrow \quad a + b = 6
\]
3. From the point \( (2, 27) \):
\[
a(2)^2 + b(2) + c = 27 \quad \Rightarrow \quad 4a + 2b + 14 = 27 \quad \Rightarrow \quad 4a + 2b = 13 \quad \Rightarrow \quad 2a + b = 6.5
\]
Now we have a system of equations:
1. \( a + b = 6 \) (Equation 1)
2. \( 2a + b = 6.5 \) (Equation 2)
Now, we can eliminate \( b \) by subtracting Equation 1 from Equation 2:
\[
(2a + b) - (a + b) = 6.5 - 6
\]
\[
a = 0.5
\]
Now, substituting \( a \) back into Equation 1 to solve for \( b \):
\[
0.5 + b = 6 \quad \Rightarrow \quad b = 5.5
\]
Now, substituting \( a \) and \( b \) back to find \( c \):
\[
c = 14
\]
Thus, the function that describes the points given is:
\[
f(x) = 0.5x^2 + 5.5x + 14
\]
It's not an arithmetic function; it is actually a quadratic function, and this results from the points provided. Please refer to the data points for the sequence, as they may not correspond directly to linear functional relationships.
To find the function based on the given points, we can first find the common difference \( m \):
1. From \( (0, 14) \) to \( (1, 20) \):
\[
m = 20 - 14 = 6
\]
2. From \( (1, 20) \) to \( (2, 27) \):
\[
m = 27 - 20 = 7
\]
It seems there’s an inconsistency here with the differences calculated. To check, let’s look at the points and determine if the arithmetic sequence is consistently applied:
- From \( f(0) = 14 \) to \( f(1) = 20 \) is indeed \( 6 \).
- From \( f(1) = 20 \) to \( f(2) = 27 \) is \( 7 \).
This inconsistency means that these points aren't forming a perfect arithmetic sequence with a uniform difference. It looks like there’s a misunderstanding of the arithmetic sequence nature.
To find a quadratic polynomial that fits these points, we will instead assume \( f(x) = ax^2 + bx + c \). Since we have three points, we can set up a system of equations using them:
1. From the point \( (0, 14) \):
\[
c = 14
\]
2. From the point \( (1, 20) \):
\[
a(1)^2 + b(1) + c = 20 \quad \Rightarrow \quad a + b + 14 = 20 \quad \Rightarrow \quad a + b = 6
\]
3. From the point \( (2, 27) \):
\[
a(2)^2 + b(2) + c = 27 \quad \Rightarrow \quad 4a + 2b + 14 = 27 \quad \Rightarrow \quad 4a + 2b = 13 \quad \Rightarrow \quad 2a + b = 6.5
\]
Now we have a system of equations:
1. \( a + b = 6 \) (Equation 1)
2. \( 2a + b = 6.5 \) (Equation 2)
Now, we can eliminate \( b \) by subtracting Equation 1 from Equation 2:
\[
(2a + b) - (a + b) = 6.5 - 6
\]
\[
a = 0.5
\]
Now, substituting \( a \) back into Equation 1 to solve for \( b \):
\[
0.5 + b = 6 \quad \Rightarrow \quad b = 5.5
\]
Now, substituting \( a \) and \( b \) back to find \( c \):
\[
c = 14
\]
Thus, the function that describes the points given is:
\[
f(x) = 0.5x^2 + 5.5x + 14
\]
It's not an arithmetic function; it is actually a quadratic function, and this results from the points provided. Please refer to the data points for the sequence, as they may not correspond directly to linear functional relationships.