To find the next three terms in the sequence, we need to first establish the rule that governs how the sequence progresses. We can start by examining the differences between consecutive terms.
From 1 to 2, the increment is \(2 - 1 = 1\).
From 2 to 4, the increment is \(4 - 2 = 2\).
From 4 to 7, the increment is \(7 - 4 = 3\).
From 7 to 11, the increment is \(11 - 7 = 4\).
From 11 to 16, the increment is \(16 - 11 = 5\).
We can observe that the differences between subsequent terms follow an arithmetic progression increasing by 1 each time.
Let's use this pattern to find the next three terms.
The last given term in the sequence is 16, and the difference before reaching 16 was 5 (from the previous term 11).
The next difference, based on our observation that each difference increases by 1, will be \(5 + 1 = 6\).
So, the next term after 16 will be:
\(16 + 6 = 22\)
Now we have our first term: 22.
The next difference will then be \(6 + 1 = 7\).
So, the term after 22 will be:
\(22 + 7 = 29\)
And that gives us our second term: 29.
Following the same pattern, the next difference will be \(7 + 1 = 8\).
So, the term after 29 will be:
\(29 + 8 = 37\)
And that provides our third term: 37.
Therefore, the next three terms in the sequence after 1, 2, 4, 7, 11, 16, are 22, 29, and 37.