The sequence is \( -1,4,-7,10, \ldots \) \( :: \) arithmetic whether the sequence is arithmetic, geometric, or neither \( :: \) a common difference of 3

To determine whether the sequence \( -1, 4, -7, 10, \ldots \) is arithmetic, geometric, or neither, we need to analyze the differences between consecutive terms. 1. **Identify the terms of the sequence:** - First term (\(a_1\)): \(-1\) - Second term (\(a_2\)): \(4\) - Third term (\(a_3\)): \(-7\) - Fourth term (\(a_4\)): \(10\) 2. **Calculate the differences between consecutive terms:** - Difference between \(a_2\) and \(a_1\): \[ d_1 = a_2 - a_1 = 4 - (-1) = 4 + 1 = 5 \] - Difference between \(a_3\) and \(a_2\): \[ d_2 = a_3 - a_2 = -7 - 4 = -11 \] - Difference between \(a_4\) and \(a_3\): \[ d_3 = a_4 - a_3 = 10 - (-7) = 10 + 7 = 17 \] 3. **Analyze the differences:** - The differences are \(5\), \(-11\), and \(17\). Since these differences are not constant, the sequence is not arithmetic. 4. **Check if the sequence is geometric:** - For a sequence to be geometric, the ratio of consecutive terms must be constant. Let's calculate the ratios: - Ratio of \(a_2\) to \(a_1\): \[ r_1 = \frac{a_2}{a_1} = \frac{4}{-1} = -4 \] - Ratio of \(a_3\) to \(a_2\): \[ r_2 = \frac{a_3}{a_2} = \frac{-7}{4} \] - Ratio of \(a_4\) to \(a_3\): \[ r_3 = \frac{a_4}{a_3} = \frac{10}{-7} \] 5. **Analyze the ratios:** - The ratios are \(-4\), \(\frac{-7}{4}\), and \(\frac{10}{-7}\). Since these ratios are not constant, the sequence is not geometric. 6. **Conclusion:** - The sequence \( -1, 4, -7, 10, \ldots \) is neither arithmetic nor geometric. The statement about a common difference of 3 is incorrect, as the differences calculated do not support that claim.