1. Problem 2: Find the next number in the sequence 98, 96, 92, 86, 78, 68, 56, ______. 2. First, find the differences between consecutive terms: $$96 - 98 = -2$$ $$92 - 96 = -4$$ $$86 - 92 = -6$$ $$78 - 86 = -8$$ $$68 - 78 = -10$$ $$56 - 68 = -12$$ 3. The differences form the sequence: -2, -4, -6, -8, -10, -12, which decreases by 2 each time. 4. The next difference should be $$-14$$. 5. Add this to the last term: $$56 + (-14) = 42$$ 6. So, the next number in sequence 2 is $$42$$. 7. Problem 3: Find the next number in the sequence 2, 8, 16, 26, 38, 52, 68, 86, 106, 128, ______. 8. Find the differences between consecutive terms: $$8 - 2 = 6$$ $$16 - 8 = 8$$ $$26 - 16 = 10$$ $$38 - 26 = 12$$ $$52 - 38 = 14$$ $$68 - 52 = 16$$ $$86 - 68 = 18$$ $$106 - 86 = 20$$ $$128 - 106 = 22$$ 9. The differences increase by 2 each time: 6, 8, 10, ..., 22. 10. The next difference should be $$24$$. 11. Add this to the last term: $$128 + 24 = 152$$ 12. So, the next number in sequence 3 is $$152$$. 13. Problem 4: Find the next number in the sequence 2, 6, -1, -3, -10, -30, -37, ______. 14. Find the differences between consecutive terms: $$6 - 2 = 4$$ $$-1 - 6 = -7$$ $$-3 - (-1) = -2$$ $$-10 - (-3) = -7$$ $$-30 - (-10) = -20$$ $$-37 - (-30) = -7$$ 15. Notice the pattern: every other difference is -7. 16. The differences alternate between -7 and other values (4, -2, -20). 17. The next difference should be $$-7$$ (following the alternating pattern). 18. Add this to the last term: $$-37 + (-7) = -44$$ 19. So, the next number in sequence 4 is $$-44$$.