Determine the missing number in the sequence: $$4, \; 11, \; 25, \; 53, \; 109, \; ?$$

Answer Key:

221

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Difference Analysis:** Successive differences are 7, 14, 28, 56 (doubling each step). 2. **Stage 2: Next Difference:** $56 \times 2 = 112$. 3. **Stage 3: Next Term:** $109 + 112 = 221$. 4. **Stage 4: Verification:** Alternatively, $T_{n} = 2 \times T_{n-1} + 3 = 2(109) + 3 = 221$. Correct answer is 221.

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