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Sagot :
To identify the explicit function for the sequence given in the table, we need to observe the pattern in the values of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 9 \\ \hline 2 & 14 \\ \hline 3 & 19 \\ \hline 4 & 24 \\ \hline 5 & 29 \\ \hline \end{tabular} \][/tex]
First, let's see the changes in the [tex]\( y \)[/tex]-values as [tex]\( x \)[/tex] increases:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 9 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 14 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 19 \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( y = 24 \)[/tex]
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 29 \)[/tex]
We notice that each time [tex]\( x \)[/tex] increases by 1, [tex]\( y \)[/tex] increases by 5. This suggests a linear relationship of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
The slope [tex]\( m \)[/tex] (or the common difference in an arithmetic sequence) is 5 because:
[tex]\[ 14 - 9 = 5, \quad 19 - 14 = 5, \quad 24 - 19 = 5, \quad 29 - 24 = 5 \][/tex]
Now we need to find the y-intercept ([tex]\( b \)[/tex]) of the linear function. For [tex]\( x = 1 \)[/tex], [tex]\( y = 9 \)[/tex]. Using the linear equation form:
[tex]\[ y = mx + b \][/tex]
[tex]\[ 9 = 5(1) + b \][/tex]
[tex]\[ 9 = 5 + b \][/tex]
[tex]\[ b = 4 \][/tex]
However, based on the structure of the options provided, we should express the function with an offset form starting from [tex]\( y \)[/tex] when [tex]\( x = 1 \)[/tex].
By observing the table carefully and analyzing the pattern:
For the sequence, the correct explicit formula is:
[tex]\[ a(n) = 9 + (n-1) \cdot 5 \][/tex]
To confirm this, we check:
- For [tex]\( x = 1 \)[/tex]: [tex]\( 9 + (1-1) \cdot 5 = 9 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( 9 + (2-1) \cdot 5 = 14 \)[/tex]
- For [tex]\( x = 3 \)[/tex]: [tex]\( 9 + (3-1) \cdot 5 = 19 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( 9 + (4-1) \cdot 5 = 24 \)[/tex]
- For [tex]\( x = 5 \)[/tex]: [tex]\( 9 + (5-1) \cdot 5 = 29 \)[/tex]
Thus, the final explicit function that matches the pattern in the table is:
B. [tex]\( a(n) = 9 + (n-1) \cdot 5 \)[/tex]
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 9 \\ \hline 2 & 14 \\ \hline 3 & 19 \\ \hline 4 & 24 \\ \hline 5 & 29 \\ \hline \end{tabular} \][/tex]
First, let's see the changes in the [tex]\( y \)[/tex]-values as [tex]\( x \)[/tex] increases:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 9 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 14 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 19 \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( y = 24 \)[/tex]
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 29 \)[/tex]
We notice that each time [tex]\( x \)[/tex] increases by 1, [tex]\( y \)[/tex] increases by 5. This suggests a linear relationship of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
The slope [tex]\( m \)[/tex] (or the common difference in an arithmetic sequence) is 5 because:
[tex]\[ 14 - 9 = 5, \quad 19 - 14 = 5, \quad 24 - 19 = 5, \quad 29 - 24 = 5 \][/tex]
Now we need to find the y-intercept ([tex]\( b \)[/tex]) of the linear function. For [tex]\( x = 1 \)[/tex], [tex]\( y = 9 \)[/tex]. Using the linear equation form:
[tex]\[ y = mx + b \][/tex]
[tex]\[ 9 = 5(1) + b \][/tex]
[tex]\[ 9 = 5 + b \][/tex]
[tex]\[ b = 4 \][/tex]
However, based on the structure of the options provided, we should express the function with an offset form starting from [tex]\( y \)[/tex] when [tex]\( x = 1 \)[/tex].
By observing the table carefully and analyzing the pattern:
For the sequence, the correct explicit formula is:
[tex]\[ a(n) = 9 + (n-1) \cdot 5 \][/tex]
To confirm this, we check:
- For [tex]\( x = 1 \)[/tex]: [tex]\( 9 + (1-1) \cdot 5 = 9 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( 9 + (2-1) \cdot 5 = 14 \)[/tex]
- For [tex]\( x = 3 \)[/tex]: [tex]\( 9 + (3-1) \cdot 5 = 19 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( 9 + (4-1) \cdot 5 = 24 \)[/tex]
- For [tex]\( x = 5 \)[/tex]: [tex]\( 9 + (5-1) \cdot 5 = 29 \)[/tex]
Thus, the final explicit function that matches the pattern in the table is:
B. [tex]\( a(n) = 9 + (n-1) \cdot 5 \)[/tex]
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