Question
Write a rule for the \( n \)th term of the sequence. Then find \( a_{2} \) \( 11.7,10.8,9.9,9, \ldots \) \( a_{n}=\square \) \( a_{20}=\square \)
Ask by Moreno Schultz. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The \( n \)th term of the sequence is \( a_n = 11.7 - 0.9(n-1) \).
\( a_{2} = 10.8 \)
\( a_{20} = -5.4 \)
Solution
- The sequence is arithmetic because each term decreases by the same constant.
- The first term is
\[
a_1 = 11.7
\]
- The common difference is
\[
d = 10.8 - 11.7 = -0.9.
\]
- The general formula for the \( n \)th term of an arithmetic sequence is
\[
a_n = a_1 + (n-1)d.
\]
- Substituting the values, we get
\[
a_n = 11.7 + (n-1)(-0.9) = 11.7 - 0.9(n-1).
\]
- To find \( a_{2} \), substitute \( n = 2 \):
\[
a_{2} = 11.7 - 0.9(2-1) = 11.7 - 0.9 = 10.8.
\]
- To find \( a_{20} \), substitute \( n = 20 \):
\[
a_{20} = 11.7 - 0.9(20-1) = 11.7 - 0.9(19) = 11.7 - 17.1 = -5.4.
\]
Thus,
\[
a_n = 11.7 - 0.9(n-1)
\]
\[
a_{2} = 10.8
\]
\[
a_{20} = -5.4
\]
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Simplify this solution Mind Expander
To find the rule for the \( n \)th term of the sequence, we notice that the terms decrease by 0.9 each time. Thus, we can express the \( n \)th term as: \[ a_{n} = 11.7 - 0.9(n-1) \] Now, to find \( a_{2} \): \[ a_{2} = 11.7 - 0.9(2-1) = 11.7 - 0.9 = 10.8 \] Next, to find \( a_{20} \): \[ a_{20} = 11.7 - 0.9(20-1) = 11.7 - 17.1 = -5.4 \] So, the answers are: \( a_{n} = 11.7 - 0.9(n-1) \) \( a_{2} = 10.8 \) \( a_{20} = -5.4 \)
