The formula [tex]S=\frac{n\left(a_1+a_n\right)}{2}[/tex] gives the partial sum of an arithmetic sequence. What is the formula solved for [tex]a_n[/tex]?

A. [tex]a_n=\frac{2 S-a_1 n}{n}[/tex]
B. [tex]a_n=\frac{2 S+a_1 n}{n}[/tex]
C. [tex]a_n=2 S+a_1 n+n[/tex]
D. [tex]a_n=2 S-a_1 n+n[/tex]



Answer :

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}
\