SOLUTION: 5,2,8,3,11,4,14,5,17,6,...

Question 1091658: 5,2,8,3,11,4,14,5,17,6,...
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Note that you can NEVER be sure of the pattern for any sequence like this, or be sure of the next few numbers in the sequence. Even if there is what looks like a logical pattern, it might not be the "right" pattern.

But a worthwhile problem like this will have a recognizable pattern which is almost certainly the right one. In this problem, there are a couple of ways you can find what is PROBABLY the way the sequence is produced.

One way is to look at the sequence as two sequences spliced together. That is, look for patterns in the alternating terms of the sequence. In this example, look at the odd-numbered terms (not the terms that are odd numbers; rather the numbers that are 1st, 3rd, 5th... in the sequence) and the even-numbered terms:
odd-numbered: 5, 8, 11, 14, 17, ...
even-numbered: 2, 3, 4, 5, 6, ...

You should be able to see a simple pattern in each of those sequences. Extending each of them a few more terms gives
5, 8, 11, 14, 17, 20, 23, ...
2, 3, 4, 5, 6, 7, 8, ...

And so we can extend the given sequence by splicing these two together:
5, 2, 8, 3, 11, 4, 14, 5, 17, 6, 20, 7, 23, 8, ...

Another way to see the pattern for producing the sequence is to look for a pattern in the differences between successive terms. With the given numbers, we see
-3, +6, -5, +8, -7, +10, -9, +12, -11, ...

This pattern is easily extended to give
-3, +6, -5, +8, -7, +10, -9, +12, -11, +14, -13, +16, -15, ...

And those differences again produce the sequence shown above.
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