To find the value of [tex]\( n \)[/tex] given the first term [tex]\( a_1 = 3 \)[/tex], the common difference [tex]\( d = 4 \)[/tex], and the [tex]\( n \)[/tex]-th term [tex]\( a_n = 59 \)[/tex] in an arithmetic sequence, we can follow these steps:
1. Understand the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
2. Substitute the known values into the formula:
[tex]\[
59 = 3 + (n-1) \cdot 4
\][/tex]
3. Isolate the term with [tex]\( n \)[/tex]:
[tex]\[
59 = 3 + 4(n-1)
\][/tex]
4. Subtract 3 from both sides of the equation to simplify:
[tex]\[
59 - 3 = 4(n-1)
\][/tex]
[tex]\[
56 = 4(n-1)
\][/tex]
5. Divide both sides by 4 to solve for [tex]\( n-1 \)[/tex]:
[tex]\[
\frac{56}{4} = n - 1
\][/tex]
[tex]\[
14 = n - 1
\][/tex]
6. Add 1 to both sides of the equation to find [tex]\( n \)[/tex]:
[tex]\[
14 + 1 = n
\][/tex]
[tex]\[
n = 15
\][/tex]
Therefore, the value of [tex]\( n \)[/tex] is [tex]\( 15 \)[/tex].
