To find the 21st term of an arithmetic sequence with the first term [tex]\(a_1 = -5\)[/tex] and common difference [tex]\(d = 6\)[/tex], we will use the formula for the n-th term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Here's the step-by-step solution:
1. Identify the given values:
- First term ([tex]\(a_1\)[/tex]): [tex]\(-5\)[/tex]
- Common difference ([tex]\(d\)[/tex]): [tex]\(6\)[/tex]
- Position ([tex]\(n\)[/tex]): [tex]\(21\)[/tex]
2. Substitute these values into the formula:
[tex]\[ a_{21} = a_1 + (21 - 1) \cdot d \][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[ a_{21} = -5 + 20 \cdot 6 \][/tex]
4. Multiply the common difference with [tex]\(20\)[/tex]:
[tex]\[ a_{21} = -5 + 120 \][/tex]
5. Add [tex]\(-5\)[/tex] to [tex]\(120\)[/tex]:
[tex]\[ a_{21} = 115 \][/tex]
Thus, the 21st term of the sequence is [tex]\(115\)[/tex].
