Sure! Let's find the expression for [tex]\( n \)[/tex] in the formula for the [tex]\( n \)[/tex]th term of an arithmetic sequence.
The formula for the [tex]\( n \)[/tex]th term of an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
We need to isolate [tex]\( n \)[/tex] in this equation. Let's start by manipulating the given equation step by step.
1. Subtract [tex]\( a_1 \)[/tex] from both sides:
[tex]\[ a_n - a_1 = (n-1)d \][/tex]
2. Divide both sides by [tex]\( d \)[/tex]:
[tex]\[ \frac{a_n - a_1}{d} = n-1 \][/tex]
3. Add 1 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[ \frac{a_n - a_1}{d} + 1 = n \][/tex]
To simplify this equation, we combine the terms on the right-hand side:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
So, the equivalent equation for [tex]\( n \)[/tex] in terms of [tex]\( a_n, a_1, \)[/tex] and [tex]\( d \)[/tex] is:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
Among the provided choices, the correct one is:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{a_n - a_1 + d}{d}} \][/tex]
