To find the [tex]\( n \)[/tex]-th term of an arithmetic sequence, we use the formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
where [tex]\( a_1 \)[/tex] is the first term, [tex]\( d \)[/tex] is the common difference, and [tex]\( n \)[/tex] is the term number we want to find.
Given the values:
[tex]\[ a_1 = 3 \][/tex]
[tex]\[ d = -8 \][/tex]
and we want to find the [tex]\( 71 \)[/tex]-st term ([tex]\( n = 71 \)[/tex]).
First, let's substitute [tex]\( a_1 \)[/tex], [tex]\( d \)[/tex], and [tex]\( n \)[/tex] into the formula:
[tex]\[ a_{71} = 3 + (71 - 1) \cdot (-8) \][/tex]
Simplify the expression step by step:
1. Calculate [tex]\( 71 - 1 \)[/tex]:
[tex]\[ 71 - 1 = 70 \][/tex]
2. Multiply by the common difference [tex]\( d \)[/tex]:
[tex]\[ 70 \cdot (-8) = -560 \][/tex]
3. Add the result to the first term [tex]\( a_1 \)[/tex]:
[tex]\[ 3 + (-560) = 3 - 560 = -557 \][/tex]
Therefore, the seventy-first term [tex]\( a_{71} \)[/tex] of the arithmetic sequence is:
[tex]\[ a_{71} = -557 \][/tex]
