To find the next three terms in the sequence 2, 5, 10, 17, 26, we first need to identify a pattern in the differences between consecutive terms.
Here are the terms of the sequence:
- 1st term: 2
- 2nd term: 5
- 3rd term: 10
- 4th term: 17
- 5th term: 26
Now, let’s calculate the differences:
- 5 – 2 = 3
- 10 – 5 = 5
- 17 – 10 = 7
- 26 – 17 = 9
This gives us a new sequence of differences: 3, 5, 7, 9.
Next, we look at the differences of these differences:
- 5 – 3 = 2
- 7 – 5 = 2
- 9 – 7 = 2
The second-level differences are constant, which indicates that the sequence is quadratic. We see that the differences are increasing by 2 each time.
Continuing this pattern, the next differences will be:
- 9 + 2 = 11
- 11 + 2 = 13
- 13 + 2 = 15
Now, we can add these new differences to the last known term to find the next terms in the sequence:
- 26 + 11 = 37
- 37 + 13 = 50
- 50 + 15 = 65
Thus, the next three terms in the sequence are: 37, 50, and 65.