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

We are given the sequence: \[ 4,\ 6,\ 14,\ 30,\ 56,\ 94,\ 146 \] **Step 1. Find the first differences.** Subtract consecutive terms: \[ 6-4=2,\quad 14-6=8,\quad 30-14=16,\quad 56-30=26,\quad 94-56=38,\quad 146-94=52 \] Thus, the first differences are: \[ 2,\ 8,\ 16,\ 26,\ 38,\ 52 \] **Step 2. Find the second differences.** Subtract consecutive first differences: \[ 8-2=6,\quad 16-8=8,\quad 26-16=10,\quad 38-26=12,\quad 52-38=14 \] So, the second differences are: \[ 6,\ 8,\ 10,\ 12,\ 14 \] **Step 3. Find the third differences.** Subtract consecutive second differences: \[ 8-6=2,\quad 10-8=2,\quad 12-10=2,\quad 14-12=2 \] The third differences are constant at: \[ 2 \] **Step 4. Use the pattern to find the next term.** Since the third differences are constant, continue the pattern: 1. The next second difference is: \[ 14 + 2 = 16 \] 2. The next first difference is: \[ 52 + 16 = 68 \] 3. The next term in the sequence is: \[ 146 + 68 = 214 \] Thus, the eighth term in the sequence is: \[ \boxed{214} \]