To analyze the given sequence:
42, 84, 126, 168, ...
We can observe that the common difference between consecutive terms is:
- \( 84 - 42 = 42 \)
- \( 126 - 84 = 42 \)
- \( 168 - 126 = 42 \)
This confirms that it is an arithmetic sequence with a common difference of 42. The first term \( a_1 \) is 42.
Let's evaluate each friend's proposed formulas:
-
Jack's formula: \( a_n = 42 \cdot 2^{(n-1)} \)
This formula suggests a geometric progression rather than an arithmetic one, as it involves multiplication by 2 raised to a power.
Let's calculate a few terms using Jack's formula:
- For \( n = 1 \): \( a_1 = 42 \cdot 2^{0} = 42 \) (correct)
- For \( n = 2 \): \( a_2 = 42 \cdot 2^{1} = 84 \) (correct)
- For \( n = 3 \): \( a_3 = 42 \cdot 2^{2} = 168 \) (incorrect, should be 126)
Jack's formula is not correct.
-
Jill's formula: \( a_n = 42 + (n-1) \cdot 42 \)
This simplifies to \( a_n = 42n \). Let's test it:
- For \( n = 1 \): \( a_1 = 42 \cdot 1 = 42 \) (correct)
- For \( n = 2 \): \( a_2 = 42 \cdot 2 = 84 \) (correct)
- For \( n = 3 \): \( a_3 = 42 \cdot 3 = 126 \) (correct)
- For \( n = 4 \): \( a_4 = 42 \cdot 4 = 168 \) (correct)
Jill's formula is correct.
-
Cinder's formula: \( a_n = 42 + (n-1) \cdot 2 \)
Let's evaluate this:
- For \( n = 1 \): \( a_1 = 42 + 0 = 42 \) (correct)
- For \( n = 2 \): \( a_2 = 42 + 1 \cdot 2 = 44 \) (incorrect)
Cinder's formula is not correct.
-
Ella's formula: \( a_1 = 42 \) and \( a_n = a_{n-1} + 42 \)
This describes the sequence recursively, starting at 42 and adding 42 for each subsequent term:
- For \( n = 1 \): \( a_1 = 42 \) (correct)
- For \( n = 2 \): \( a_2 = a_{1} + 42 = 42 + 42 = 84 \) (correct)
- For \( n = 3 \): \( a_3 = a_{2} + 42 = 84 + 42 = 126 \) (correct)
- For \( n = 4 \): \( a_4 = a_{3} + 42 = 126 + 42 = 168 \) (correct)
Ella's formula is also correct.
Conclusion:
Jill's and Ella's formulas are correct for generating the sequence. Jack's and Cinder's formulas do not accurately represent the arithmetic sequence.