To identify which given equation is equivalent to the [tex]\( n \)[/tex]-th term formula of an arithmetic sequence, let's start with the standard formula for the [tex]\( n \)[/tex]-th term:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
where:
- [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term of the sequence,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference,
- [tex]\( n \)[/tex] is the term number.
We need to rearrange the formula to solve for [tex]\( n \)[/tex]:
Firstly, isolate the term involving [tex]\( n \)[/tex]:
[tex]\[ a_n - a_1 = (n - 1)d \][/tex]
Next, divide both sides by [tex]\( d \)[/tex] to isolate [tex]\( n - 1 \)[/tex]:
[tex]\[ \frac{a_n - a_1}{d} = n - 1 \][/tex]
Now, add 1 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{a_n - a_1}{d} + 1 \][/tex]
Rearranging it to enhance readability:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
So, the equivalent equation to the given formula [tex]\( a_n = a_1 + (n - 1)d \)[/tex] is:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
Upon examining the given options:
- [tex]\( n = a_n + a_1 \)[/tex]
- [tex]\( n = \frac{a_n + a_1 - d}{d} \)[/tex]
- [tex]\( n = a_n - a_1 \)[/tex]
- [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]
Thus, the correct answer is:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
