Answer :
Sure, let's find the 81st term of the arithmetic sequence [tex]\(-10, -25, -40, \ldots\)[/tex].
1. Identify the first term and the common difference:
- The first term ([tex]\(a_1\)[/tex]) of the sequence is [tex]\(-10\)[/tex].
- The common difference ([tex]\(d\)[/tex]) is the difference between any two consecutive terms. Thus, [tex]\(d\)[/tex] can be calculated as:
[tex]\[
d = -25 - (-10) = -25 + 10 = -15
\][/tex]
2. Use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
The general formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Here, we need to find the 81st term, so [tex]\(n = 81\)[/tex].
3. Substitute the known values into the formula:
- First term ([tex]\(a_1\)[/tex]) = [tex]\(-10\)[/tex]
- Common difference ([tex]\(d\)[/tex]) = [tex]\(-15\)[/tex]
- Term number ([tex]\(n\)[/tex]) = 81
So, we substitute these values into the formula:
[tex]\[
a_{81} = -10 + (81 - 1) \cdot (-15)
\][/tex]
4. Perform the calculations:
- Calculate [tex]\(81 - 1\)[/tex]:
[tex]\[
81 - 1 = 80
\][/tex]
- Multiply this by the common difference [tex]\(d = -15\)[/tex]:
[tex]\[
80 \cdot (-15) = -1200
\][/tex]
- Add this result to the first term [tex]\(a_1 = -10\)[/tex]:
[tex]\[
a_{81} = -10 + (-1200) = -10 - 1200 = -1210
\][/tex]
Therefore, the 81st term of the arithmetic sequence [tex]\(-10, -25, -40, \ldots\)[/tex] is [tex]\(-1210\)[/tex].
1. Identify the first term and the common difference:
- The first term ([tex]\(a_1\)[/tex]) of the sequence is [tex]\(-10\)[/tex].
- The common difference ([tex]\(d\)[/tex]) is the difference between any two consecutive terms. Thus, [tex]\(d\)[/tex] can be calculated as:
[tex]\[
d = -25 - (-10) = -25 + 10 = -15
\][/tex]
2. Use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
The general formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Here, we need to find the 81st term, so [tex]\(n = 81\)[/tex].
3. Substitute the known values into the formula:
- First term ([tex]\(a_1\)[/tex]) = [tex]\(-10\)[/tex]
- Common difference ([tex]\(d\)[/tex]) = [tex]\(-15\)[/tex]
- Term number ([tex]\(n\)[/tex]) = 81
So, we substitute these values into the formula:
[tex]\[
a_{81} = -10 + (81 - 1) \cdot (-15)
\][/tex]
4. Perform the calculations:
- Calculate [tex]\(81 - 1\)[/tex]:
[tex]\[
81 - 1 = 80
\][/tex]
- Multiply this by the common difference [tex]\(d = -15\)[/tex]:
[tex]\[
80 \cdot (-15) = -1200
\][/tex]
- Add this result to the first term [tex]\(a_1 = -10\)[/tex]:
[tex]\[
a_{81} = -10 + (-1200) = -10 - 1200 = -1210
\][/tex]
Therefore, the 81st term of the arithmetic sequence [tex]\(-10, -25, -40, \ldots\)[/tex] is [tex]\(-1210\)[/tex].
