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Answer :
Certainly! Let's find the 81st term of the arithmetic sequence [tex]\(-10, -25, -40, \ldots\)[/tex].
1. Identify the first term [tex]\( a \)[/tex] and the common difference [tex]\( d \)[/tex]
- The first term [tex]\( a \)[/tex] is the first number in the sequence, which is [tex]\(-10\)[/tex].
- To find the common difference [tex]\( d \)[/tex], subtract 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 for the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] of an arithmetic sequence is:
[tex]\[
a_n = a + (n - 1) \cdot d
\][/tex]
- Here, we need to find the 81st term, so [tex]\( n = 81 \)[/tex].
3. Substitute the values of [tex]\( a \)[/tex], [tex]\( d \)[/tex], and [tex]\( n \)[/tex] into the formula
- [tex]\( a = -10 \)[/tex]
- [tex]\( d = -15 \)[/tex]
- [tex]\( n = 81 \)[/tex]
[tex]\[
a_{81} = -10 + (81 - 1) \cdot (-15)
\][/tex]
4. Calculate the expression step-by-step
- First, simplify inside the parentheses:
[tex]\[
81 - 1 = 80
\][/tex]
- Then, multiply by the common difference [tex]\( d \)[/tex]:
[tex]\[
80 \cdot (-15) = -1200
\][/tex]
- Finally, add this result to the first term:
[tex]\[
-10 + (-1200) = -10 - 1200 = -1210
\][/tex]
Therefore, the 81st term of the arithmetic sequence [tex]\(-10, -25, -40, \ldots\)[/tex] is [tex]\(-1210\)[/tex].
1. Identify the first term [tex]\( a \)[/tex] and the common difference [tex]\( d \)[/tex]
- The first term [tex]\( a \)[/tex] is the first number in the sequence, which is [tex]\(-10\)[/tex].
- To find the common difference [tex]\( d \)[/tex], subtract 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 for the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] of an arithmetic sequence is:
[tex]\[
a_n = a + (n - 1) \cdot d
\][/tex]
- Here, we need to find the 81st term, so [tex]\( n = 81 \)[/tex].
3. Substitute the values of [tex]\( a \)[/tex], [tex]\( d \)[/tex], and [tex]\( n \)[/tex] into the formula
- [tex]\( a = -10 \)[/tex]
- [tex]\( d = -15 \)[/tex]
- [tex]\( n = 81 \)[/tex]
[tex]\[
a_{81} = -10 + (81 - 1) \cdot (-15)
\][/tex]
4. Calculate the expression step-by-step
- First, simplify inside the parentheses:
[tex]\[
81 - 1 = 80
\][/tex]
- Then, multiply by the common difference [tex]\( d \)[/tex]:
[tex]\[
80 \cdot (-15) = -1200
\][/tex]
- Finally, add this result to the first term:
[tex]\[
-10 + (-1200) = -10 - 1200 = -1210
\][/tex]
Therefore, the 81st term of the arithmetic sequence [tex]\(-10, -25, -40, \ldots\)[/tex] is [tex]\(-1210\)[/tex].
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