Describe the rule for each number sequence in your own words and determine the next three t of the sequence: 1. \( 1 ; 3 ; 7 ; 15 ; \ldots \) 2. \( 70 ; 80 ; 100 ; 140 ; \ldots \) 3. \( 16 ; 14 ; 10 ; 2 ; \ldots \) \( 2 ; 5 ; 10 ; 17 ; \ldots \) \( 3 ; 5 ; 9 ; 15 ; \ldots \) IONS AND ALGEBRA UNIT 6: NUMERIC AND GEOMETRIC PATTERNS

The first sequence is generated by doubling the previous number and then subtracting one, which gives us 1, 3 (1*2 + 1), 7 (3*2 + 1), 15 (7*2 + 1). So, the next three terms would be 31 (15*2 + 1), 63 (31*2 + 1), and 127 (63*2 + 1). The second sequence increases in steps that grow larger. It begins at 70, jumps by 10 to reach 80, then leaps 20 to get to 100, and finally hops 40 to arrive at 140. The next terms will be 200 (adding 60), 280 (adding 80), and 380 (adding 100). The third sequence moves downward quickly, starting at 16 and dropping to 14 (a decrease of 2), then a much more significant decline to 10 (a decrease of 4), and finally plunging to 2 (a decrease of 8). The pattern shows us that we’re doubling the previous decrease of 2, so the next terms will be -6 (2 - 8) and then -20 (-6 - 14). The sequence with numbers 2, 5, 10, and 17 appears to be generated by adding consecutive odd numbers starting from 3: 2 + 3 = 5, 5 + 5 = 10, 10 + 7 = 17. Continuing this trend, we’ll add 9 and 11 respectively to obtain the next terms 26 and 37. In the fifth sequence of numbers, the pattern starts from 3 and goes to 5 (increase by 2), then 5 to 9 (5 + 4), and then 9 to 15 (a further +6). The increments are increasing by 2 in their respective jumps, so the next jumps will be +8 and +10, giving us the next terms: 24 (15 + 9) and 39 (24 + 15).