Question
Chinedu sat down to do his homework, which included
\[30\] math problems. He solved
\[2\] problems each minute.
Let
\[f(n)\] be the number of problems left for Chinedu to solve at the beginning of the
\[n^\text{th}\] minute.
\[f\] is a sequence. What kind of sequence is it?
Choose 1 answer:
Choose 1 answer:
(Choice A) Arithmetic sequence
A
Arithmetic sequence
(Choice B) Geometric sequence
B
Geometric sequence
Write an explicit formula for the sequence.
\[f(n)=\]
\[30\] math problems. He solved
\[2\] problems each minute.
Let
\[f(n)\] be the number of problems left for Chinedu to solve at the beginning of the
\[n^\text{th}\] minute.
\[f\] is a sequence. What kind of sequence is it?
Choose 1 answer:
Choose 1 answer:
(Choice A) Arithmetic sequence
A
Arithmetic sequence
(Choice B) Geometric sequence
B
Geometric sequence
Write an explicit formula for the sequence.
\[f(n)=\]
Answers
GPT-4o mini
To analyze the situation, we start with the initial number of math problems, which is \(30\). Chinedu solves \(2\) problems each minute.
The function \(f(n)\) represents the number of problems left at the beginning of the \(n^\text{th}\) minute. Therefore:
- At \(n = 0\) (the beginning), he has \(f(0) = 30\) problems.
- At \(n = 1\), he has solved \(2\) problems, so \(f(1) = 30 - 2 \times 1 = 28\).
- At \(n = 2\), he has solved \(4\) problems, so \(f(2) = 30 - 2 \times 2 = 26\).
- At \(n = 3\), he has solved \(6\) problems, so \(f(3) = 30 - 2 \times 3 = 24\).
- And so on.
From the above, we can see that with each passing minute, the number of problems left decreases by \(2\). This shows that the sequence is linear, characterized by a constant difference.
The general formula of an arithmetic sequence can be written as:
\[
f(n) = f(0) - d \times n
\]
where:
- \(f(0)\) is the initial term,
- \(d\) is the common difference (the amount decreased each term).
In this case:
- \(f(0) = 30\),
- \(d = 2\).
Thus, the explicit formula for the sequence is:
\[
f(n) = 30 - 2n
\]
### Choose the answer:
The type of sequence is (Choice A) Arithmetic sequence.
### Explicit Formula:
\[
f(n) = 30 - 2n
\]
The function \(f(n)\) represents the number of problems left at the beginning of the \(n^\text{th}\) minute. Therefore:
- At \(n = 0\) (the beginning), he has \(f(0) = 30\) problems.
- At \(n = 1\), he has solved \(2\) problems, so \(f(1) = 30 - 2 \times 1 = 28\).
- At \(n = 2\), he has solved \(4\) problems, so \(f(2) = 30 - 2 \times 2 = 26\).
- At \(n = 3\), he has solved \(6\) problems, so \(f(3) = 30 - 2 \times 3 = 24\).
- And so on.
From the above, we can see that with each passing minute, the number of problems left decreases by \(2\). This shows that the sequence is linear, characterized by a constant difference.
The general formula of an arithmetic sequence can be written as:
\[
f(n) = f(0) - d \times n
\]
where:
- \(f(0)\) is the initial term,
- \(d\) is the common difference (the amount decreased each term).
In this case:
- \(f(0) = 30\),
- \(d = 2\).
Thus, the explicit formula for the sequence is:
\[
f(n) = 30 - 2n
\]
### Choose the answer:
The type of sequence is (Choice A) Arithmetic sequence.
### Explicit Formula:
\[
f(n) = 30 - 2n
\]