The question you opened
Seven, ten, sixteen, twenty eight
A sequence starts 7, 10, 16, 28.
What is the next number in the sequence?
52
The gaps are 3, 6, 12, doubling each time. The next gap is 24, and 28 plus 24 gives 52.
The sequence runs 7, 10, 16, 28, and the next term is 52. The move that cracks almost every number sequence is to stop looking at the numbers and start looking at the gaps between them. From 7 to 10 is 3. From 10 to 16 is 6. From 16 to 28 is 12. Written out, the gaps are 3, 6, 12, and those are doubling. So the next gap is 24, and 28 plus 24 gives 52.
Carry on and the gaps become 48, 96, 192, producing 100, 196 and 388. There is a second way to see the same thing, and it is worth having because it gives you a formula. Each term is a little more than double the one before: 10 is 2 times 7 minus 4, 16 is 2 times 10 minus 4, 28 is 2 times 16 minus 4. So the rule is double and subtract 4.
Applying it to 28 gives 56 minus 4, which is 52 again. The two descriptions agree because subtracting a constant each time is what makes the differences double cleanly. Wanting a formula also tells you something about these puzzles in general. Any finite list of numbers can be continued in infinitely many ways that are all technically valid, so a sequence question is really asking for the simplest rule that fits, and simple usually means constant differences, constant ratios, or one of those applied to the differences.
That gives a short checklist you can run in order. First, are the differences constant. Second, are the ratios constant. Third, are the differences themselves doubling or following a pattern. Fourth, is each term built from the previous one by a fixed multiply and add. This sequence fails the first two tests and passes the third and fourth. Once you know to take differences, it takes about five seconds, and the same first step will solve most sequences you meet.
Technique: Take differences, then look again