To solve for [tex]\( a_n \)[/tex] from the given formula [tex]\( S = \frac{n(a_1 + a_n)}{2} \)[/tex], we will proceed step by step:
1. Start with the original formula:
[tex]\[
S = \frac{n(a_1 + a_n)}{2}
\][/tex]
2. Multiply both sides by 2 to eliminate the fraction:
[tex]\[
2S = n(a_1 + a_n)
\][/tex]
3. Divide both sides by [tex]\( n \)[/tex] to isolate [tex]\( a_1 + a_n \)[/tex]:
[tex]\[
\frac{2S}{n} = a_1 + a_n
\][/tex]
4. Subtract [tex]\( a_1 \)[/tex] from both sides to solve for [tex]\( a_n \)[/tex]:
[tex]\[
a_n = \frac{2S}{n} - a_1
\][/tex]
5. Simplify the expression:
[tex]\[
a_n = \frac{2S - a_1 n}{n}
\][/tex]
Thus, the formula solved for [tex]\( a_n \)[/tex] is:
[tex]\[
a_n = \frac{2S - a_1 n}{n}
\][/tex]
Hence, the correct option is:
\[
a_n = \frac{2 S - a_1 n}{n}
\
