Answer :
To find the 81st term of the arithmetic sequence [tex]\(-10, -25, -40, \ldots\)[/tex], we can follow these steps:
1. Identify the first term ([tex]\(a_1\)[/tex]) and the common difference ([tex]\(d\)[/tex]):
- The first term, [tex]\(a_1\)[/tex], of the sequence is [tex]\(-10\)[/tex].
- The common difference, [tex]\(d\)[/tex], is found by subtracting the first term from the second term:
[tex]\[
d = -25 - (-10) = -25 + 10 = -15
\][/tex]
2. Use the formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence:
The formula to find the [tex]\(n\)[/tex]th term ([tex]\(a_n\)[/tex]) in an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Here, [tex]\(a_1\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the term number.
3. Plug in the values to find the 81st term:
- The term number [tex]\(n\)[/tex] is 81.
- The first term [tex]\(a_1\)[/tex] is [tex]\(-10\)[/tex].
- The common difference [tex]\(d\)[/tex] is [tex]\(-15\)[/tex].
Substituting these values into the formula:
[tex]\[
a_{81} = -10 + (81 - 1) \cdot (-15)
\][/tex]
Simplify inside the parentheses:
[tex]\[
a_{81} = -10 + 80 \cdot (-15)
\][/tex]
Multiply:
[tex]\[
a_{81} = -10 + (-1200)
\][/tex]
Add the numbers:
[tex]\[
a_{81} = -10 - 1200 = -1210
\][/tex]
Therefore, the 81st term of the arithmetic sequence [tex]\(-10, -25, -40, \ldots\)[/tex] is [tex]\(-1210\)[/tex].
1. Identify the first term ([tex]\(a_1\)[/tex]) and the common difference ([tex]\(d\)[/tex]):
- The first term, [tex]\(a_1\)[/tex], of the sequence is [tex]\(-10\)[/tex].
- The common difference, [tex]\(d\)[/tex], is found by subtracting the first term from the second term:
[tex]\[
d = -25 - (-10) = -25 + 10 = -15
\][/tex]
2. Use the formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence:
The formula to find the [tex]\(n\)[/tex]th term ([tex]\(a_n\)[/tex]) in an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Here, [tex]\(a_1\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the term number.
3. Plug in the values to find the 81st term:
- The term number [tex]\(n\)[/tex] is 81.
- The first term [tex]\(a_1\)[/tex] is [tex]\(-10\)[/tex].
- The common difference [tex]\(d\)[/tex] is [tex]\(-15\)[/tex].
Substituting these values into the formula:
[tex]\[
a_{81} = -10 + (81 - 1) \cdot (-15)
\][/tex]
Simplify inside the parentheses:
[tex]\[
a_{81} = -10 + 80 \cdot (-15)
\][/tex]
Multiply:
[tex]\[
a_{81} = -10 + (-1200)
\][/tex]
Add the numbers:
[tex]\[
a_{81} = -10 - 1200 = -1210
\][/tex]
Therefore, the 81st term of the arithmetic sequence [tex]\(-10, -25, -40, \ldots\)[/tex] is [tex]\(-1210\)[/tex].
