Need help finding the rule to this pattern: 3, 4, 3, 4, 5, 3, 4, 5, _, _, _, _, _, 7

What I've thought of so far

This is all I have gotten so far. I saw that the difference between the 1st term and second term is +1 then the 2nd term and 3rd term is is -1, 4th term to 5th is +1 but then 5th term to 6th term is -2. 6th term to 7th term is +1. 7th to 8th is +1 again. I've tried where I multiplied the term number by a number and added or subtracted but I can't figure out where this pattern is coming from.
 
I need some guidance on the following sequence

For the sequence, 3,4,3,4,5,3,4,5,_,_,_,_,_,7 I have tried finding the difference between the term numbers but none have added up to get 7 for the 15th number. please help
 
It seems to me like all of your work so far has been focused solely on trying to find a common difference between the terms, to which end you've discovered there doesn't appear to be any such pattern. There's nothing wrong with this in and of itself, but it is very very limited thinking, and so it's no wonder you haven't found an answer. You need to broaden your horizons and "think outside the box," as it were. Sequences defined by common differences, called arithmetic sequences, are just one type of sequence, but there are countless others. Another common type of sequence is a geometric sequence, where each term is a constant multiple times the previous. Let's see if that holds here...

3 is the first term, so that's our base. 4 = 1.33... * 3, and 3 = 0.75 * 4. After that, we're back to 4 = 1.33... * 3. Looks good so far, maybe this is the pattern? Oh, but then 5 = 1.25 * 4, and 3 = 5 * 0.6. No dice there. So we can conclude that the sequence isn't geometric either.

This type of trial-and-error method is a great starting out method, if nothing else because it quickly eliminates types of sequence it's not and gives you guidance on how best to hone your thinking. For this particular sequence, one possible answer immediately jumps out to me just from looking at subsequences. Take only the first two terms (3 and 4). What do you notice about them? Now take only the next three terms (3, 4, and 5). What do you notice about them? Now let's look at the final three given terms (3, 4, and 5 again). What do you notice about them? Are you noticing a meta-level pattern within these subsequences? What number should come next, to continue this pattern? Finally, can you generate a subsequence of five numbers that again continues the pattern? Does this fit with the fact that 7 is given as the 14th member of the overall sequence?
 
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