To derive the equivalent equation for the [tex]\( n \)[/tex]-th term of an arithmetic sequence using the formula [tex]\( a_n = a_1 + (n - 1)d \)[/tex], we need to solve for [tex]\( n \)[/tex]. Let's go through the steps to isolate [tex]\( n \)[/tex] on one side of the equation.
Starting with the given formula:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
Step 1: Subtract [tex]\( a_1 \)[/tex] from both sides:
[tex]\[ a_n - a_1 = (n - 1)d \][/tex]
Step 2: Divide both sides by [tex]\( d \)[/tex]:
[tex]\[ \frac{a_n - a_1}{d} = n - 1 \][/tex]
Step 3: Add 1 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[ \frac{a_n - a_1}{d} + 1 = n \][/tex]
We can rewrite this equation for clarity:
[tex]\[ n = \frac{a_n - a_1}{d} + 1 \][/tex]
To combine the fraction, we can express 1 as [tex]\( \frac{d}{d} \)[/tex] and add it to the fraction:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
So the correct equivalent equation is:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
From the given choices:
1. [tex]\( n = a_n + a_1 \)[/tex]
2. [tex]\( n = \frac{a_n + a_1 - d}{d} \)[/tex]
3. [tex]\( n = a_n - a_1 \)[/tex]
4. [tex]\( n = \frac{a_n - a_1 + d}{d} \)[/tex]
The correct equivalent equation is:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
Therefore, the answer is the fourth option:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
