1. **Stating the problem:** We are given a sequence of numbers and need to find the pattern or rule governing the sequence. 2. **Analyzing the first sequence:** 4, 21, 12, 36, 108, 324 (with a note \(\times 3\) under 108 and 324). 3. **Check the pattern:** - From 4 to 21: multiply by \(\frac{21}{4} = 5.25\) - From 21 to 12: multiply by \(\frac{12}{21} = \frac{4}{7}\) - From 12 to 36: multiply by 3 - From 36 to 108: multiply by 3 - From 108 to 324: multiply by 3 4. The last three steps multiply by 3, confirming the note. 5. **Conclusion:** The sequence does not have a simple constant multiplier but from 12 onwards it multiplies by 3 each time. 6. **Analyzing the second sequence:** 5, 10, 7, 14, 11, 22, 19, 38 with note (-3, \(\times 2\)) 7. Check pattern: - 5 to 10: \(\times 2\) - 10 to 7: \(-3\) - 7 to 14: \(\times 2\) - 14 to 11: \(-3\) - 11 to 22: \(\times 2\) - 22 to 19: \(-3\) - 19 to 38: \(\times 2\) 8. Pattern alternates between multiplying by 2 and subtracting 3. 9. **Summary:** The first sequence multiplies by 3 from the third term onward. The second sequence alternates between multiplying by 2 and subtracting 3. Final answer for the first sequence pattern: from the third term onward, multiply by 3. Final answer for the second sequence pattern: alternate between multiplying by 2 and subtracting 3.