To find \( a_{57} \) for the sequence given, we first need to identify a pattern in the sequence:
The sequence is: \[ a_1 = -5, \quad a_2 = -\frac{9}{2}, \quad a_3 = -3, \quad a_4 = -\frac{3}{2}, \quad a_5 = 0, \ldots \]
Let's express this in simpler terms:
- \( a_1 = -5 = -\frac{10}{2} \)
- \( a_2 = -\frac{9}{2} \)
- \( a_3 = -3 = -\frac{6}{2} \)
- \( a_4 = -\frac{3}{2} \)
- \( a_5 = 0 = 0 \)
Now, we can look for a pattern in terms of \( n \).
Calculating the values: \[ \begin{align*} a_1 & = -5 \ a_2 & = -\frac{9}{2} \ a_3 & = -3 \ a_4 & = -\frac{3}{2} \ a_5 & = 0 \ \end{align*} \]
If we observe the fractions, they seem to go up by \( \frac{3}{2} \) for each step:
- From \( a_1 = -5 \) to \( a_2 = -\frac{9}{2} \): increase of \( \frac{1}{2} \)
- From \( a_2 = -\frac{9}{2} \) to \( a_3 = -3 \): increase of \( \frac{3}{2} \)
- From \( a_3 = -3 \) to \( a_4 = -\frac{3}{2} \): increase of \( \frac{3}{2} \)
- From \( a_4 = -\frac{3}{2} \) to \( a_5 = 0 \): increase of \( \frac{3}{2} \)
The pattern suggests that the \( n \)-th term can be modeled as: \[ a_n = -5 + \frac{3}{2}(n-1) \]
To find \( a_{57} \): \[ a_{57} = -5 + \frac{3}{2}(57-1) \] \[ = -5 + \frac{3}{2}(56) \] \[ = -5 + \frac{168}{2} \] \[ = -5 + 84 \] \[ = 84 - 5 = 79 \]
It appears I made a mistake in simplifying the calculations. Let's re-evaluate that.
\[ a_{57} = -5 + \frac{3}{2}(56) = -5 + 84 = 79 \]
Those computations adjust based on the value that \( -10 + \frac{3}{2}(n-1) \): \[ = -5 + 3(n-1)/2 = -5 + (3n-3)/2 = (-10 + 3n - 3)/2= (3n - 13)/2 \]
So we can find \( a_{57} \): \[ = (3(57) - 13)/2 = (171 - 13)/2 = 159/2 \]
Therefore, the answer to \( a_{57} \) is: \[ \boxed{\frac{159}{2}} \]