To find the 30th term of the given arithmetic sequence, we'll use the formula for the nth term of an arithmetic sequence, which is:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Where:
- [tex]\( a_n \)[/tex] is the nth term we want to find.
- [tex]\( a_1 \)[/tex] is the first term of the sequence.
- [tex]\( d \)[/tex] is the common difference.
- [tex]\( n \)[/tex] is the term number.
Given values:
- The first term [tex]\( a_1 \)[/tex] is [tex]\(-20\)[/tex].
- The common difference [tex]\( d \)[/tex] is the difference between any two successive terms in the sequence, which is [tex]\( -16 - (-20) = -16 + 20 = 4 \)[/tex].
- We want to find the 30th term, so [tex]\( n = 30 \)[/tex].
Step-by-Step Solution:
1. Start with the general formula for the nth term:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
2. Substitute the known values into the formula:
[tex]\[ a_{30} = -20 + (30-1) \cdot 4 \][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[ a_{30} = -20 + 29 \cdot 4 \][/tex]
4. Multiply 29 by 4:
[tex]\[ 29 \cdot 4 = 116 \][/tex]
5. Add the result to -20:
[tex]\[ a_{30} = -20 + 116 = 96 \][/tex]
Thus, the 30th term of the sequence is:
[tex]\[ a_{30} = 96 \][/tex]
