To find the value of [tex]\( n \)[/tex] from the given formula for the [tex]\( n \)[/tex]th term of an arithmetic sequence, let's start with the equation:
[tex]\[ a_n = a_1 + (n-1) \cdot d \][/tex]
We'll rearrange this equation to express [tex]\( n \)[/tex] in terms of [tex]\( a_n \)[/tex], [tex]\( a_1 \)[/tex], and [tex]\( d \)[/tex].
1. Begin with the initial formula:
[tex]\[ a_n = a_1 + (n-1) \cdot d \][/tex]
2. Subtract [tex]\( a_1 \)[/tex] from both sides to isolate the term involving [tex]\( n \)[/tex]:
[tex]\[ a_n - a_1 = (n-1) \cdot d \][/tex]
3. Divide both sides of the equation by [tex]\( d \)[/tex] to further isolate [tex]\( n-1 \)[/tex]:
[tex]\[ \frac{a_n - a_1}{d} = n - 1 \][/tex]
4. Add 1 to both sides of the equation to solve for [tex]\( n \)[/tex]:
[tex]\[ \frac{a_n - a_1}{d} + 1 = n \][/tex]
Finally, the equation we derived for [tex]\( n \)[/tex] is:
[tex]\[ n = \frac{a_n - a_1}{d} + 1 \][/tex]
Now, let's compare this derived formula to the given options:
1. [tex]\( n = a_n + a_1 \)[/tex]: This is not equivalent to our derived formula.
2. [tex]\( n = \frac{a_n + a_1 - d}{d} \)[/tex]: This is not equivalent to our derived formula.
3. [tex]\( n = a_n - a_1 \)[/tex]: This is not equivalent to our derived formula.
4. [tex]\( n = \frac{a_n - a_1 + d}{d} \)[/tex]: This can be rewritten as:
[tex]\[ \frac{a_n - a_1 + d}{d} = \frac{a_n - a_1}{d} + \frac{d}{d} \][/tex]
[tex]\[ = \frac{a_n - a_1}{d} + 1 \][/tex]
This matches our derived formula exactly.
Therefore, the correct option equivalent to the initial equation is:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
