Problem 1....
I will ignore the last number in the sequence you show, since it is almost certainly a typo.
(By the way, it would be polite of you to make sure the problem is correct before you post it....)
So I will use the first five of the terms you show and use those to find the correct 6th term and the new 7th term.
17 15 25 53 105
-2 10 28 52
12 18 24
6 6
The row of third differences is constant. To find two more terms in the sequence, add two more 6's in that row and work back up the array.
17 15 25 53 105 187 305
-2 10 28 52 82 118
12 18 24 30 36
6 6 6 6
ANSWERS: the correct 6th term is 187; the 7th term is 305.
Problem 2....
a1=2; a2=4; a3=6; a4=8; a5=10...
Clearly the nth term is 2n; a30 = 60
Problem 3....
(Note your statement of the problem is not complete....)
The numbers of regions for 1, 2, 3, and 4 cuts are
2, 4, 7, 11
Compare this sequence to the sequence of triangular numbers:
1, 3, 6, 10
The numbers in this sequence are each 1 more than the corresponding triangular number.
The formula for the n-th triangular number is
The formula for the maximum number of pieces of pizza with n cuts is