Question

Find the first term of the arithmetic sequence in which a subscript 27 baseline equals negative Start Fraction 1 over 2 End Fraction and the common difference is 2.
(1 point)
Responses

Start Fraction 103 over 2 End Fraction
Image with alt text: Start Fraction 103 over 2 End Fraction

negative Start Fraction 105 over 2 End Fraction
Image with alt text: negative Start Fraction 105 over 2 End Fraction

negative Start Fraction 103 over 2 End Fraction
Image with alt text: negative Start Fraction 103 over 2 End Fraction

Start Fraction 111 over 2 End Fraction

Answer 1

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:

\[
\boxed{-\frac{105}{2}}
\]