To determine which formula is correct, we first need to analyze the given sequence: 26, 10, -6, -22.
- Identify the pattern:
- The first term \( g(1) \) is 26.
- The second term \( g(2) \) is 10.
- The third term \( g(3) \) is -6.
- The fourth term \( g(4) \) is -22.
We can calculate the differences between consecutive terms:
- \( 10 - 26 = -16 \)
- \( -6 - 10 = -16 \)
- \( -22 - (-6) = -16 \)
The common difference is -16, indicating that the sequence is arithmetic.
- General formula for an arithmetic sequence: The general formula for an arithmetic sequence can be given by: \[ g(n) = g(1) + (n - 1)d \] where \( g(1) \) is the first term and \( d \) is the common difference.
In our case:
- \( g(1) = 26 \)
- \( d = -16 \)
So, substituting these values into the formula, we have: \[ g(n) = 26 + (n - 1)(-16) \]
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Simplifying the formula: \[ g(n) = 26 - 16(n - 1) \] This matches Vera's formula, \( g(n) = 26 - 16(n - 1) \).
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Verifying Fenyang's formula: Now let's check Fenyang's formula: \( g(n) = 42 - 16n \). To find \( g(1) \): \[ g(1) = 42 - 16(1) = 42 - 16 = 26 \] This part works, but we need to see if it generates the correct values for subsequent terms:
- For \( n = 2 \): \[ g(2) = 42 - 16(2) = 42 - 32 = 10 \]
- For \( n = 3 \): \[ g(3) = 42 - 16(3) = 42 - 48 = -6 \]
- For \( n = 4 \): \[ g(4) = 42 - 16(4) = 42 - 64 = -22 \]
Both formulas give the same terms, but Vera's formula is the explicit form based on the definition of arithmetic sequences. It highlights the common difference clearly.
- Conclusion: Both formulas yield the correct terms for the sequence, but Vera's is the more standard representation of the arithmetic sequence.
Thus, Vera's formula is correct.