To find the first term of the arithmetic sequence given \( a_{27} = -\frac{1}{2} \) and the common difference \( d = 2 \), we can use the formula for the \( n \)-th term of an arithmetic sequence:
\[ a_n = a_1 + (n - 1)d \]
Where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.
Given:
- \( n = 27 \)
- \( a_{27} = -\frac{1}{2} \)
- \( d = 2 \)
We can substitute these values into the formula:
\[ -\frac{1}{2} = a_1 + (27 - 1) \cdot 2 \]
This simplifies to:
\[ -\frac{1}{2} = a_1 + 26 \cdot 2 \] \[ -\frac{1}{2} = a_1 + 52 \]
Now, to isolate \( a_1 \), subtract 52 from both sides:
\[ a_1 = -\frac{1}{2} - 52 \]
We can convert \( 52 \) to a fraction with a denominator of 2:
\[ 52 = \frac{104}{2} \] \[ a_1 = -\frac{1}{2} - \frac{104}{2} = -\frac{1 + 104}{2} = -\frac{105}{2} \]
Thus, the first term \( a_1 \) is:
\[ \boxed{-\frac{105}{2}} \]