Question
Noa makes $48000 in her1 st year working,$56000 in her2 nd year,$64000 in her 3rd year, and so on. Assume the pattern continues. How much will Noa make in her th year working?
Answers
GPT-4o mini
Noa's earnings increase by $8,000 each year. We can express her earnings for each year using the following pattern:
- Year 1: $48,000
- Year 2: $56,000
- Year 3: $64,000
We can notice that the earnings form an arithmetic sequence where:
- First term (\(a\)) = $48,000
- Common difference (\(d\)) = $8,000
The formula for the \(n\)-th term of an arithmetic sequence is given by:
\[
a_n = a + (n-1) \cdot d
\]
Substituting \(a\) and \(d\):
\[
a_n = 48000 + (n-1) \cdot 8000
\]
Simplifying this:
\[
a_n = 48000 + 8000(n-1)
\]
\[
a_n = 48000 + 8000n - 8000
\]
\[
a_n = 40000 + 8000n
\]
So, Noa's earnings in her \(n\)-th year will be:
\[
a_n = 8000n + 40000
\]
Therefore, Noa will make \(8000n + 40000\) in her \(n\)-th year working.
- Year 1: $48,000
- Year 2: $56,000
- Year 3: $64,000
We can notice that the earnings form an arithmetic sequence where:
- First term (\(a\)) = $48,000
- Common difference (\(d\)) = $8,000
The formula for the \(n\)-th term of an arithmetic sequence is given by:
\[
a_n = a + (n-1) \cdot d
\]
Substituting \(a\) and \(d\):
\[
a_n = 48000 + (n-1) \cdot 8000
\]
Simplifying this:
\[
a_n = 48000 + 8000(n-1)
\]
\[
a_n = 48000 + 8000n - 8000
\]
\[
a_n = 40000 + 8000n
\]
So, Noa's earnings in her \(n\)-th year will be:
\[
a_n = 8000n + 40000
\]
Therefore, Noa will make \(8000n + 40000\) in her \(n\)-th year working.