Let's start from the given formula for the partial sum of an arithmetic sequence:
[tex]\[ S = \frac{n(a_1 + a_n)}{2} \][/tex]
We need to solve for [tex]\( a_n \)[/tex]. We can do this step-by-step by isolating [tex]\( a_n \)[/tex] on one side of the equation.
### Step 1: Multiply both sides by 2
First, eliminate the fraction by multiplying both sides of the equation by 2:
[tex]\[ 2S = n(a_1 + a_n) \][/tex]
### Step 2: Divide both sides by [tex]\( n \)[/tex]
Now, divide both sides by [tex]\( n \)[/tex] to isolate the term [tex]\( a_1 + a_n \)[/tex]:
[tex]\[ \frac{2S}{n} = a_1 + a_n \][/tex]
### Step 3: Subtract [tex]\( a_1 \)[/tex] from both sides
To solve for [tex]\( a_n \)[/tex], subtract [tex]\( a_1 \)[/tex] from both sides of the equation:
[tex]\[ a_n = \frac{2S}{n} - a_1 \][/tex]
So the formula for [tex]\( a_n \)[/tex] is:
[tex]\[ a_n = \frac{2S}{n} - a_1 \][/tex]
#### Verification with Given Options:
Given options:
1. [tex]\( a_n = \frac{2 S - a_1 n}{n} \)[/tex]
2. [tex]\( a_n = \frac{2 S + a_1 n}{n} \)[/tex]
3. [tex]\( a_n = 2 S + a_1 n + n \)[/tex]
4. [tex]\( a_n = 2 S - a_1 n + n \)[/tex]
By comparing our derived formula [tex]\(\frac{2S}{n} - a_1\)[/tex] with the options, we see that the correct option is:
[tex]\[ a_n = \frac{2S - a_1 n}{n} \][/tex]
Thus, the correct formula is:
[tex]\[ a_n = \frac{2S - a_1 n}{n} \][/tex]
