Find the pattern and fill in the missing entries. In (x) out (y)
23
2 8
-2 -4
8 26
0
302
1.5
Rule:
can you please explain to me what work you have done?
56
To find the pattern and fill in the missing entries, we need to look for a relationship between the input (x) and the output (y). In this case, we have the given inputs and outputs:
In (x) | Out (y)
--------|--------
23 |
2 | 8
-2 | -4
8 | 26
0 |
| 302
1.5 |
Looking at the values, we can see that there is some arithmetic operation happening between the input and output. Let's examine the differences between the input and output:
In (x) | Out (y) | Difference (y - x)
--------|--------|---------------------
23 | |
2 | 8 | 6
-2 | -4 | -2
8 | 26 | 18
0 | |
| 302 |
1.5 | |
From this, we can see that the difference between the input (x) and output (y) is not constant. However, if we look at the difference between the differences, we might find a pattern:
In (x) | Out (y) | Difference (y - x) | Difference (y - x) - (y - x) (previous row)
--------|--------|---------------------|---------------------------------------------------
23 | |
2 | 8 | 6 |
-2 | -4 | -2 | -4
8 | 26 | 18 | 20
0 | | |
| 302 | |
1.5 | | |
From this second table, we can see that the difference between the differences is not constant either. However, if we examine the differences between the differences, we might find a pattern:
In (x) | Out (y) | Difference (y - x) | Difference (y - x) - (y - x) (previous row) | Difference (y - x) - (y - x) - (previous row) (previous row)
--------|--------|---------------------|---------------------------------------------------|-----------------------------------------------------------------------
23 | |
2 | 8 | 6 | |
-2 | -4 | -2 | -4 |
8 | 26 | 18 | 20 | 16
0 | | | |
| 302 | | |
1.5 | | | |
From this third table, we can see that the difference between the differences of the differences is constant. Therefore, we can establish a pattern:
In (x) | Out (y)
--------|-------
23 | 25
2 | 8
-2 | -4
8 | 26
0 | 202
1.5 | 227
The pattern is as follows:
1. Take the difference between the input (x) and output (y).
2. Take the difference between the differences obtained in step 1.
3. Keep applying step 2 until a constant difference is obtained.
4. Add the constant difference obtained in step 3 to the previous value of the output (y) to get the next value.
Using this pattern, we can fill in the missing entries:
- For the input (x) of 0, the difference between the input and output is 202 (compared to 8 and -4).
- For the input (x) of 1.5, the difference between the input and output is 227 (compared to 26 and -4).
Therefore, the missing entries in the table are:
In (x) | Out (y)
--------|-------
0 | 202
1.5 | 227