To find the explicit formula for the sequence \(-51, -34, -17, 0, \ldots\), we need to determine the pattern in the given terms.
First, let's identify the first term and the common difference:
- The first term \(h(1) = -51\).
- The second term \(h(2) = -34\).
- The third term \(h(3) = -17\).
- The fourth term \(h(4) = 0\).
To determine the common difference, we subtract the first term from the second term: \[ -34 - (-51) = -34 + 51 = 17 \] So, the common difference \(d\) is \(17\).
Now, we can describe the \(n\)-th term of an arithmetic sequence using the formula: \[ h(n) = h(1) + (n-1) \cdot d \] Substituting the values we have: \[ h(n) = -51 + (n-1) \cdot 17 \] Simplifying the formula: \[ h(n) = -51 + 17n - 17 \] \[ h(n) = 17n - 68 \]
Now, let's compare this with Oliver's and Putri's formulas:
- Oliver's formula: \(h(n) = -51 + 17n\)
- Putri's formula: \(h(n) = -68 + 17n\)
We can see that Oliver's formula simplifies to: \[ h(n) = 17n - 51 \]
This is incorrect because it does not match our derived formula.
On the other hand, Putri's formula: \[ h(n) = -68 + 17n \] can be rearranged to \(h(n) = 17n - 68\), which matches our derived formula.
Therefore, Putri is correct. The correct explicit formula for the sequence is: \[ h(n) = -68 + 17n \]