To determine which of the given formulas is equivalent to the formula for the [tex]\( n \)[/tex]th term of an arithmetic sequence, we start with the given formula:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
We need to solve for [tex]\( n \)[/tex]. Let's start by rearranging the terms step-by-step.
1. Subtract [tex]\( a_1 \)[/tex] from both sides to isolate the term involving [tex]\( n \)[/tex]:
[tex]\[ a_n - a_1 = (n-1)d \][/tex]
2. Next, divide both sides by [tex]\( d \)[/tex] to solve for [tex]\( n-1 \)[/tex]:
[tex]\[ \frac{a_n - a_1}{d} = n-1 \][/tex]
3. Finally, add 1 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{a_n - a_1}{d} + 1 \][/tex]
Now, let's compare this to the given options:
1. [tex]\( n = a_n + a_1 \)[/tex]
- This option does not match our derived formula.
2. [tex]\( n = \frac{a_n + a_1 - d}{d} \)[/tex]
- This option also does not match our derived formula.
3. [tex]\( n = a_n - a_1 \)[/tex]
- This option is not correct as it lacks [tex]\( d \)[/tex] and does not match our derived formula.
4. [tex]\( n = \frac{a_n - a_1 + d}{d} \)[/tex]
- To see if this matches, we can rewrite our derived formula slightly:
[tex]\[ n = \frac{a_n - a_1 + d - d}{d} + 1 \][/tex]
[tex]\[ n = \frac{a_n - a_1 + d}{d} - \frac{d}{d} + 1 \][/tex]
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
This simplified form [tex]\( n = \frac{a_n - a_1 + d}{d} \)[/tex] matches our derived result closely. Thus, the correct equivalent formula is:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
So, the equivalent formula is given in option 4:
[tex]\[ n = \frac{a_n - a_1 + d}{d} \][/tex]
