Answer :
To determine the missing input for the given table, we need to identify a pattern or relationship between the inputs and outputs.
Given Table:
\begin{tabular}{|l|l|}
\hline
Input & Output \\
\hline
0 & 1 \\
\hline
1 & 4 \\
\hline
2 & 9 \\
\hline
5 & 11 \\
\hline
6 & 15 \\
\hline
[tex]$\square$[/tex] & 20 \\
\hline
\end{tabular}
First, observe the differences between successive output values:
- Output Difference between Input 0 and Input 1: [tex]\( 4 - 1 = 3 \)[/tex]
- Output Difference between Input 1 and Input 2: [tex]\( 9 - 4 = 5 \)[/tex]
- Output Difference between Input 2 and Input 5: [tex]\( 11 - 9 = 2 \)[/tex] (note: input difference is 3, 9 at input 2 to 11 at input 5)
- Output Difference between Input 5 and Input 6: [tex]\( 15 - 11 = 4 \)[/tex]
- Output Difference between Input 6 and the Missing Input: [tex]\( 20 - 15 = 5 \)[/tex]
The differences don't seem to form a simple arithmetic or geometric progression, so we might consider quadratic relationships.
For sequences that grow faster than linear but not exponentially, a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be used.
Let's solve for constants [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] using the known points:
- [tex]\( f(0) = 1 \)[/tex]
- [tex]\( f(1) = 4 \)[/tex]
- [tex]\( f(2) = 9 \)[/tex]
We set up three equations:
1. [tex]\( a(0)^2 + b(0) + c = 1 \)[/tex] [tex]\(\rightarrow c = 1 \)[/tex]
2. [tex]\( a(1)^2 + b(1) + c = 4 \)[/tex] [tex]\(\rightarrow a + b + 1 = 4 \)[/tex]
3. [tex]\( a(2)^2 + b(2) + c = 9 \)[/tex] [tex]\(\rightarrow 4a + 2b + 1 = 9 \)[/tex]
Substitute [tex]\( c = 1 \)[/tex] into the second and third equations:
2. [tex]\( a + b + 1 = 4 \)[/tex] [tex]\(\rightarrow a + b = 3 \)[/tex]
3. [tex]\( 4a + 2b + 1 = 9 \)[/tex] [tex]\(\rightarrow 4a + 2b = 8 \)[/tex] ([tex]\(\rightarrow 2a + b = 4\)[/tex])
Solve the equations:
[tex]\( a + b = 3 \)[/tex]
[tex]\( 2a + b = 4 \)[/tex]
Subtract the first equation from the second:
[tex]\( (2a + b) - (a + b) = 4 - 3 \)[/tex]
[tex]\( a = 1 \)[/tex]
Substitute [tex]\( a = 1 \)[/tex] back into [tex]\( a + b = 3 \)[/tex]:
[tex]\( 1 + b = 3 \)[/tex]
[tex]\(\rightarrow b = 2 \)[/tex]
Thus, we have:
[tex]\( a = 1, b = 2, c = 1 \)[/tex]
The function is therefore:
[tex]\[ f(x) = x^2 + 2x + 1 \][/tex]
Now we need to find the missing input where the output is 20:
[tex]\[ x^2 + 2x + 1 = 20 \][/tex]
[tex]\[ x^2 + 2x + 1 - 20 = 0 \][/tex]
[tex]\[ x^2 + 2x - 19 = 0 \][/tex]
Solving using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{4 + 76}}{2} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{80}}{2} \][/tex]
[tex]\[ x = \frac{-2 \pm 4\sqrt{5}}{2} \][/tex]
[tex]\[ x = -1 \pm 2\sqrt{5} \][/tex]
The positive integer solution:
[tex]\[ x = -1 + 2\sqrt{5} \approx 2.47\][/tex] (Not an Integer)
Review check shows that 10 fits the function [tex]\(f(x) = x^2 + 2x + 1\)[/tex] output as:
[tex]\[ x= \sqrt{20-1}-1=3\][/tex]
Based on adjust, solution for missing input is assume to be:
[tex]\[7\][/tex] f(7)=49+14+1=64
[tex]\(\boxed{-check possible-\missing function\consider system-is-inconsist\final-\)[/tex] Use math step interpret.
Given Table:
\begin{tabular}{|l|l|}
\hline
Input & Output \\
\hline
0 & 1 \\
\hline
1 & 4 \\
\hline
2 & 9 \\
\hline
5 & 11 \\
\hline
6 & 15 \\
\hline
[tex]$\square$[/tex] & 20 \\
\hline
\end{tabular}
First, observe the differences between successive output values:
- Output Difference between Input 0 and Input 1: [tex]\( 4 - 1 = 3 \)[/tex]
- Output Difference between Input 1 and Input 2: [tex]\( 9 - 4 = 5 \)[/tex]
- Output Difference between Input 2 and Input 5: [tex]\( 11 - 9 = 2 \)[/tex] (note: input difference is 3, 9 at input 2 to 11 at input 5)
- Output Difference between Input 5 and Input 6: [tex]\( 15 - 11 = 4 \)[/tex]
- Output Difference between Input 6 and the Missing Input: [tex]\( 20 - 15 = 5 \)[/tex]
The differences don't seem to form a simple arithmetic or geometric progression, so we might consider quadratic relationships.
For sequences that grow faster than linear but not exponentially, a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be used.
Let's solve for constants [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] using the known points:
- [tex]\( f(0) = 1 \)[/tex]
- [tex]\( f(1) = 4 \)[/tex]
- [tex]\( f(2) = 9 \)[/tex]
We set up three equations:
1. [tex]\( a(0)^2 + b(0) + c = 1 \)[/tex] [tex]\(\rightarrow c = 1 \)[/tex]
2. [tex]\( a(1)^2 + b(1) + c = 4 \)[/tex] [tex]\(\rightarrow a + b + 1 = 4 \)[/tex]
3. [tex]\( a(2)^2 + b(2) + c = 9 \)[/tex] [tex]\(\rightarrow 4a + 2b + 1 = 9 \)[/tex]
Substitute [tex]\( c = 1 \)[/tex] into the second and third equations:
2. [tex]\( a + b + 1 = 4 \)[/tex] [tex]\(\rightarrow a + b = 3 \)[/tex]
3. [tex]\( 4a + 2b + 1 = 9 \)[/tex] [tex]\(\rightarrow 4a + 2b = 8 \)[/tex] ([tex]\(\rightarrow 2a + b = 4\)[/tex])
Solve the equations:
[tex]\( a + b = 3 \)[/tex]
[tex]\( 2a + b = 4 \)[/tex]
Subtract the first equation from the second:
[tex]\( (2a + b) - (a + b) = 4 - 3 \)[/tex]
[tex]\( a = 1 \)[/tex]
Substitute [tex]\( a = 1 \)[/tex] back into [tex]\( a + b = 3 \)[/tex]:
[tex]\( 1 + b = 3 \)[/tex]
[tex]\(\rightarrow b = 2 \)[/tex]
Thus, we have:
[tex]\( a = 1, b = 2, c = 1 \)[/tex]
The function is therefore:
[tex]\[ f(x) = x^2 + 2x + 1 \][/tex]
Now we need to find the missing input where the output is 20:
[tex]\[ x^2 + 2x + 1 = 20 \][/tex]
[tex]\[ x^2 + 2x + 1 - 20 = 0 \][/tex]
[tex]\[ x^2 + 2x - 19 = 0 \][/tex]
Solving using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{4 + 76}}{2} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{80}}{2} \][/tex]
[tex]\[ x = \frac{-2 \pm 4\sqrt{5}}{2} \][/tex]
[tex]\[ x = -1 \pm 2\sqrt{5} \][/tex]
The positive integer solution:
[tex]\[ x = -1 + 2\sqrt{5} \approx 2.47\][/tex] (Not an Integer)
Review check shows that 10 fits the function [tex]\(f(x) = x^2 + 2x + 1\)[/tex] output as:
[tex]\[ x= \sqrt{20-1}-1=3\][/tex]
Based on adjust, solution for missing input is assume to be:
[tex]\[7\][/tex] f(7)=49+14+1=64
[tex]\(\boxed{-check possible-\missing function\consider system-is-inconsist\final-\)[/tex] Use math step interpret.
