Certainly! Let's analyze and decode the pattern in the given table.
### Observing the Table
The table presented is:
| First Column (i) | Second Column (Value) |
|------------------|------------------------|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
| 6 | 64 |
### Identifying the Pattern
To find the relationship between the first column (i) and the second column (Value):
1. Look at the sequence in the second column:
- 1, 2, 4, 8, 16, 32, 64
2. Notice if there's any multiplication or exponential relationship:
- 1 (2^0)
- 2 (2^1)
- 4 (2^2)
- 8 (2^3)
- 16 (2^4)
- 32 (2^5)
- 64 (2^6)
### Derivation
Each value in the second column seems to be a power of 2, where the exponent is the corresponding value from the first column.
- [tex]\( V(0) = 2^0 = 1 \)[/tex]
- [tex]\( V(1) = 2^1 = 2 \)[/tex]
- [tex]\( V(2) = 2^2 = 4 \)[/tex]
- [tex]\( V(3) = 2^3 = 8 \)[/tex]
- [tex]\( V(4) = 2^4 = 16 \)[/tex]
- [tex]\( V(5) = 2^5 = 32 \)[/tex]
- [tex]\( V(6) = 2^6 = 64 \)[/tex]
Thus, the second column values follow the function [tex]\( V(i) = 2^i \)[/tex].
### Conclusion
By observing the pattern from the table, we determined that the values in the second column are powers of 2, relative to their row index (starting from 0). Hence, the relationship can be summarized as:
[tex]\[ \text{Second Column Value} = 2^{\text{First Column Value}} \][/tex]
So, the table is:
| i | [tex]\(2^i\)[/tex] |
|---|--------|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
| 6 | 64 |
This matches perfectly with the given table data.
