1. **Stating the problem:** We have 10 rows, each with a sequence of three numbers followed by nine empty squares. We need to find the next nine numbers in each sequence. 2. **Understanding the problem:** Each row represents a number pattern sequence. Our goal is to identify the pattern in the first three numbers and extend it by nine more numbers. 3. **Approach:** For each row, analyze the difference or ratio between the first three numbers to identify if the sequence is arithmetic (constant difference), geometric (constant ratio), or another pattern. 4. **Row 1:** Numbers: 117, 109, 101 - Differences: 109 - 117 = -8, 101 - 109 = -8 - Pattern: Arithmetic sequence with common difference $d = -8$ - Next nine numbers: $$101 - 8 = 93$$ $$93 - 8 = 85$$ $$85 - 8 = 77$$ $$77 - 8 = 69$$ $$69 - 8 = 61$$ $$61 - 8 = 53$$ $$53 - 8 = 45$$ $$45 - 8 = 37$$ $$37 - 8 = 29$$ 5. **Row 2:** Numbers: 12, 112, 212 - Differences: 112 - 12 = 100, 212 - 112 = 100 - Pattern: Arithmetic sequence with $d = 100$ - Next nine numbers: $$212 + 100 = 312$$ $$312 + 100 = 412$$ $$412 + 100 = 512$$ $$512 + 100 = 612$$ $$612 + 100 = 712$$ $$712 + 100 = 812$$ $$812 + 100 = 912$$ $$912 + 100 = 1012$$ $$1012 + 100 = 1112$$ 6. **Row 3:** Numbers: 156, 149, 142 - Differences: 149 - 156 = -7, 142 - 149 = -7 - Pattern: Arithmetic with $d = -7$ - Next nine numbers: $$142 - 7 = 135$$ $$135 - 7 = 128$$ $$128 - 7 = 121$$ $$121 - 7 = 114$$ $$114 - 7 = 107$$ $$107 - 7 = 100$$ $$100 - 7 = 93$$ $$93 - 7 = 86$$ $$86 - 7 = 79$$ 7. **Row 4:** Numbers: 88, 100, 112 - Differences: 100 - 88 = 12, 112 - 100 = 12 - Pattern: Arithmetic with $d = 12$ - Next nine numbers: $$112 + 12 = 124$$ $$124 + 12 = 136$$ $$136 + 12 = 148$$ $$148 + 12 = 160$$ $$160 + 12 = 172$$ $$172 + 12 = 184$$ $$184 + 12 = 196$$ $$196 + 12 = 208$$ $$208 + 12 = 220$$ 8. **Row 5:** Numbers: 58, 55, 52 - Differences: 55 - 58 = -3, 52 - 55 = -3 - Pattern: Arithmetic with $d = -3$ - Next nine numbers: $$52 - 3 = 49$$ $$49 - 3 = 46$$ $$46 - 3 = 43$$ $$43 - 3 = 40$$ $$40 - 3 = 37$$ $$37 - 3 = 34$$ $$34 - 3 = 31$$ $$31 - 3 = 28$$ $$28 - 3 = 25$$ 9. **Row 6:** Numbers: 25, 75, 125 - Differences: 75 - 25 = 50, 125 - 75 = 50 - Pattern: Arithmetic with $d = 50$ - Next nine numbers: $$125 + 50 = 175$$ $$175 + 50 = 225$$ $$225 + 50 = 275$$ $$275 + 50 = 325$$ $$325 + 50 = 375$$ $$375 + 50 = 425$$ $$425 + 50 = 475$$ $$475 + 50 = 525$$ $$525 + 50 = 575$$ 10. **Row 7:** Numbers: 121, 116, 111 - Differences: 116 - 121 = -5, 111 - 116 = -5 - Pattern: Arithmetic with $d = -5$ - Next nine numbers: $$111 - 5 = 106$$ $$106 - 5 = 101$$ $$101 - 5 = 96$$ $$96 - 5 = 91$$ $$91 - 5 = 86$$ $$86 - 5 = 81$$ $$81 - 5 = 76$$ $$76 - 5 = 71$$ $$71 - 5 = 66$$ 11. **Row 8:** Numbers: 69, 73, 77 - Differences: 73 - 69 = 4, 77 - 73 = 4 - Pattern: Arithmetic with $d = 4$ - Next nine numbers: $$77 + 4 = 81$$ $$81 + 4 = 85$$ $$85 + 4 = 89$$ $$89 + 4 = 93$$ $$93 + 4 = 97$$ $$97 + 4 = 101$$ $$101 + 4 = 105$$ $$105 + 4 = 109$$ $$109 + 4 = 113$$ 12. **Row 9:** Numbers: 233, 208, 183 - Differences: 208 - 233 = -25, 183 - 208 = -25 - Pattern: Arithmetic with $d = -25$ - Next nine numbers: $$183 - 25 = 158$$ $$158 - 25 = 133$$ $$133 - 25 = 108$$ $$108 - 25 = 83$$ $$83 - 25 = 58$$ $$58 - 25 = 33$$ $$33 - 25 = 8$$ $$8 - 25 = -17$$ $$-17 - 25 = -42$$ 13. **Row 10:** Numbers: 53, 62, 71 - Differences: 62 - 53 = 9, 71 - 62 = 9 - Pattern: Arithmetic with $d = 9$ - Next nine numbers: $$71 + 9 = 80$$ $$80 + 9 = 89$$ $$89 + 9 = 98$$ $$98 + 9 = 107$$ $$107 + 9 = 116$$ $$116 + 9 = 125$$ $$125 + 9 = 134$$ $$134 + 9 = 143$$ $$143 + 9 = 152$$ **Final answers:** - Row 1: 93, 85, 77, 69, 61, 53, 45, 37, 29 - Row 2: 312, 412, 512, 612, 712, 812, 912, 1012, 1112 - Row 3: 135, 128, 121, 114, 107, 100, 93, 86, 79 - Row 4: 124, 136, 148, 160, 172, 184, 196, 208, 220 - Row 5: 49, 46, 43, 40, 37, 34, 31, 28, 25 - Row 6: 175, 225, 275, 325, 375, 425, 475, 525, 575 - Row 7: 106, 101, 96, 91, 86, 81, 76, 71, 66 - Row 8: 81, 85, 89, 93, 97, 101, 105, 109, 113 - Row 9: 158, 133, 108, 83, 58, 33, 8, -17, -42 - Row 10: 80, 89, 98, 107, 116, 125, 134, 143, 152