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

GinnyAnswer
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]

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