Final answer:
The next number in the sequence is calculated using the method of successive differences, which reveals a pattern where each difference between numbers increases by 1. Adding the next difference of 13 to the last term 57 gives us 70.
Step-by-step explanation:
To determine the next number in the sequence using the method of successive differences, we first look at the differences between consecutive terms in the sequence given: 0, 7, 15, 24, 34, 45, 57. Let's calculate this step by step:
- 7 - 0 = 7
- 15 - 7 = 8
- 24 - 15 = 9
- 34 - 24 = 10
- 45 - 34 = 11
- 57 - 45 = 12
The differences are increasing by 1 each time. To continue the pattern, the next difference should be 13. Therefore, to find the next number in the sequence, we add 13 to the last term:
57 + 13 = 70
So, the next number in the sequence is 70.