Question
Challenge problem
7*) Select all the formulas that correctly represent the arithmetic sequence
\[101, 114, 127,\unicode{0x2026}\]
Choose all answers that apply:
Choose all answers that apply:
(Choice A)
\[\begin{cases}
j(1)=13\\\\
j(n)=j(n-1)+101
\end{cases}\]
A
\[\begin{cases}
j(1)=13\\\\
j(n)=j(n-1)+101
\end{cases}\]
(Choice B)
\[\begin{cases}
j(1)=101\\\\
j(n)=j(n-1)+13
\end{cases}\]
B
\[\begin{cases}
j(1)=101\\\\
j(n)=j(n-1)+13
\end{cases}\]
(Choice C)
\[\begin{cases}
j(1)=114\\\\
j(n)=j(n-1)+13
\end{cases}\]
C
\[\begin{cases}
j(1)=114\\\\
j(n)=j(n-1)+13
\end{cases}\]
(Choice D)
\[j(n)=101+13n\]
D
\[j(n)=101+13n\]
(Choice E)
\[j(n)=88+13n\]
E
\[j(n)=88+13n\]
(Choice F)
\[j(n)=114+13n\]
F
\[j(n)=114+13n\]
(Choice G)
\[j(n)=101+13(n-1)\]
G
\[j(n)=101+13(n-1)\]
7*) Select all the formulas that correctly represent the arithmetic sequence
\[101, 114, 127,\unicode{0x2026}\]
Choose all answers that apply:
Choose all answers that apply:
(Choice A)
\[\begin{cases}
j(1)=13\\\\
j(n)=j(n-1)+101
\end{cases}\]
A
\[\begin{cases}
j(1)=13\\\\
j(n)=j(n-1)+101
\end{cases}\]
(Choice B)
\[\begin{cases}
j(1)=101\\\\
j(n)=j(n-1)+13
\end{cases}\]
B
\[\begin{cases}
j(1)=101\\\\
j(n)=j(n-1)+13
\end{cases}\]
(Choice C)
\[\begin{cases}
j(1)=114\\\\
j(n)=j(n-1)+13
\end{cases}\]
C
\[\begin{cases}
j(1)=114\\\\
j(n)=j(n-1)+13
\end{cases}\]
(Choice D)
\[j(n)=101+13n\]
D
\[j(n)=101+13n\]
(Choice E)
\[j(n)=88+13n\]
E
\[j(n)=88+13n\]
(Choice F)
\[j(n)=114+13n\]
F
\[j(n)=114+13n\]
(Choice G)
\[j(n)=101+13(n-1)\]
G
\[j(n)=101+13(n-1)\]
Answers
GPT-4o mini
To determine which of the given choices correctly represents the arithmetic sequence \(101, 114, 127, \ldots\), we need to first identify the parameters of the sequence:
**First term (a)**: \(a = 101\)
**Common difference (d)**: The difference between consecutive terms is \(d = 114 - 101 = 13\) (and likewise, \(127 - 114 = 13\)).
The general formula for the n-th term \(j(n)\) of an arithmetic sequence can be expressed as:
\[
j(n) = a + (n-1)d
\]
Substituting our values:
\[
j(n) = 101 + (n-1) \cdot 13
\]
This can also be written as:
\[
j(n) = 101 + 13(n-1)
\]
This identifies Choice G as valid.
Next, we can also express the n-th term in a different way:
\[
j(n) = 101 + 13n - 13 = 101 + 13n - 13 = 88 + 13n
\]
Thus, we find that Choice E is also valid.
Now, let's evaluate each choice provided:
- **Choice A**: Incorrect.
\(j(1) = 13\) is not correct.
- **Choice B**: Correct.
\(j(1) = 101\) and since \(j(n) = j(n-1) + 13\) correctly describes the sequence.
- **Choice C**: Incorrect.
\(j(1) = 114\) does not reflect the first term of the sequence.
- **Choice D**: Incorrect.
The formula \(j(n) = 101 + 13n\) represents the wrong adjustment for n; it should either represent \(j(n) = 101 + 13(n-1)\) or we would see a shift in sequence.
- **Choice E**: Correct.
\(j(n) = 88 + 13n\) is valid, as shown above.
- **Choice F**: Incorrect.
\(j(n) = 114 + 13n\) does not accurately represent the sequence.
- **Choice G**: Correct.
\(j(n) = 101 + 13(n-1)\) matches our established formula.
In conclusion, the valid choices representing the arithmetic sequence are:
- **Choice B**
- **Choice E**
- **Choice G**
**First term (a)**: \(a = 101\)
**Common difference (d)**: The difference between consecutive terms is \(d = 114 - 101 = 13\) (and likewise, \(127 - 114 = 13\)).
The general formula for the n-th term \(j(n)\) of an arithmetic sequence can be expressed as:
\[
j(n) = a + (n-1)d
\]
Substituting our values:
\[
j(n) = 101 + (n-1) \cdot 13
\]
This can also be written as:
\[
j(n) = 101 + 13(n-1)
\]
This identifies Choice G as valid.
Next, we can also express the n-th term in a different way:
\[
j(n) = 101 + 13n - 13 = 101 + 13n - 13 = 88 + 13n
\]
Thus, we find that Choice E is also valid.
Now, let's evaluate each choice provided:
- **Choice A**: Incorrect.
\(j(1) = 13\) is not correct.
- **Choice B**: Correct.
\(j(1) = 101\) and since \(j(n) = j(n-1) + 13\) correctly describes the sequence.
- **Choice C**: Incorrect.
\(j(1) = 114\) does not reflect the first term of the sequence.
- **Choice D**: Incorrect.
The formula \(j(n) = 101 + 13n\) represents the wrong adjustment for n; it should either represent \(j(n) = 101 + 13(n-1)\) or we would see a shift in sequence.
- **Choice E**: Correct.
\(j(n) = 88 + 13n\) is valid, as shown above.
- **Choice F**: Incorrect.
\(j(n) = 114 + 13n\) does not accurately represent the sequence.
- **Choice G**: Correct.
\(j(n) = 101 + 13(n-1)\) matches our established formula.
In conclusion, the valid choices representing the arithmetic sequence are:
- **Choice B**
- **Choice E**
- **Choice G**