1, 6, 8, 3, 7, ...

Crypto hint: REBMU NTIGI DEERH TASTI

Reading the 'crypto hint' in reverse, we can see that we need a three digit number as our answer.  But how does that help us?  In fact, it doesn't really, to begin with!  The clue is in the problem title; in particular, in the word EXTEND.  Consider the numbers that start this sequence... more to the point, consider how they are spelled.  Perhaps you can see a pattern?  Again, use the title as your guide - 'Extend, at the end'.  Think about it now, before you read the rest of my solution.

Well, the numbers given are 1, 6, 8, 3, and 7.  If we spell them, we have one, six, eight, three, seven.  Look at the ENDS of these words - onE, siX, eighT, threE, seveN, ...  The final letters are E,X,T,E and N!  So now the pattern has become clear.  We are spelling out the word EXTEND with the final letters of the numbers in our sequence, and all we need to do is find a number that ends in the required D.  Now we can use the hint given originally - we need a 3 digit number.  100 (one hundreD) will work, although so will 200, 300, 400, ..., 900.  (In fact, I suppose that any number with three or more digits that ended in xy00 would also work, where the digits x and y are not both zero.)

So, I believe that the intended solution is the number 100.  A tricky problem!  (Usually I can't work these out to save myself)