Use differences to find a pattern in the sequence. \( 4,6,14,30,56,94,146 \) Assuming that the pattern continues, the eighth term should be

To find a pattern in the sequence \( 4, 6, 14, 30, 56, 94, 146 \), let’s calculate the differences between consecutive terms: 1. \( 6 - 4 = 2 \) 2. \( 14 - 6 = 8 \) 3. \( 30 - 14 = 16 \) 4. \( 56 - 30 = 26 \) 5. \( 94 - 56 = 38 \) 6. \( 146 - 94 = 52 \) So, the first differences are \( 2, 8, 16, 26, 38, 52 \). Now, let's find the second differences: 1. \( 8 - 2 = 6 \) 2. \( 16 - 8 = 8 \) 3. \( 26 - 16 = 10 \) 4. \( 38 - 26 = 12 \) 5. \( 52 - 38 = 14 \) The second differences are \( 6, 8, 10, 12, 14 \). The pattern in the second differences increases by \( 2 \) each time. Continuing this, the next second difference would be \( 16 \). If we add that to the last first difference \( 52 \): \[ 52 + 16 = 68 \] Now, we add this to the last term in the original sequence: \[ 146 + 68 = 214 \] Thus, the eighth term should be \( 214 \).