Question

Make a conjecture about the next item in each sequence.

1) 5, –10, 15, –20
2) –2, 1, −12, 14, −18
3) 12, 6, 3, 1.5, 0.75
4) –4, –1, 2, 5, 8
5) –2, 4, –8, 16, –32

Answer 1

The conjectured next items in the sequences are 25, 33, 0.375, 11, and 64, following the identified patterns of alternating signs, tripled and reduced values, halving, incrementing by three, and alternating signs with doubling, respectively.

Step-by-step explanation:

To make a conjecture about the next item in each sequence, we need to identify the pattern that each sequence follows.

1. The sequence 5, –10, 15, –20 increases by 5 each time but alternates in sign (+, –, +, –). The next term would be positive and 5 more than 20, which is 25.
2. The sequence –2, 1, –12, 14, –18 doesn't have a clear arithmetical pattern. However, a possible pattern could be that each positive number is followed by a negative number that is 9 less than the triple of the previous positive number. Following this pattern, the next term would be 3 times 14 minus 9, which is 33.
3. The sequence 12, 6, 3, 1.5, 0.75 is each number being halved (÷ 2). The next term would be half of 0.75, which is 0.375.
4. The sequence –4, –1, 2, 5, 8 increases by 3 each time. The next term would be 3 more than 8, which is 11.
5. The sequence –2, 4, –8, 16, –32 alternates signs and each term is double the absolute value of the previous term. The next term would be positive and double of 32, which is 64.