To solve the formula \( S = \frac{n (a_1 + a_n)}{2} \) for \( a_n \), we will follow a detailed, step-by-step mathematical approach:
1. Understand the given formula:
The formula provided is:
[tex]\[
S = \frac{n (a_1 + a_n)}{2}
\][/tex]
where \( S \) is the partial sum of an arithmetic sequence, \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the nth term.
2. Isolate the term involving \( a_n \):
First, we need to eliminate the fraction. Multiply both sides of the equation by 2:
[tex]\[
2S = n (a_1 + a_n)
\][/tex]
3. Solve for \( a_n \):
Divide both sides of the equation by \( n \) to isolate \( a_1 + a_n \):
[tex]\[
\frac{2S}{n} = a_1 + a_n
\][/tex]
4. Remove \( a_1 \) from the equation:
To isolate \( a_n \), subtract \( a_1 \) from both sides of the equation:
[tex]\[
\frac{2S}{n} - a_1 = a_n
\][/tex]
Therefore, the formula for \( a_n \) is:
[tex]\[
a_n = \frac{2S}{n} - a_1
\][/tex]
None of the provided options match our derived formula exactly. Thus, the correct formula for solving \( a_n \) from the given sum formula is:
[tex]\[
a_n = \frac{2S}{n} - a_1
\][/tex]
