Maybe a fuller explanation is warranted.
Statement 1 gives us the definition of the sequence. i here represents the term number, so the second term is x2, the third term is x3, the nth term is xn, etc. i > 1 here means "For each term beyond the first". This is an immediate sign that Statement 1 is INSUFFICIENT: if our definition doesn't work for the term we're supposed to find, we won't be able to find that term! (Another reason is that we aren't given any terms in the sequence.)
Statement 2 gives us an algebraic equation -- NOT a definition -- relating x5 to x4. This isn't helpful on its own, as we don't know how to relate any other terms of the sequence together.
Taking the two statements together, we have TWO WAYS of relating x4 and x5 to each other. The first way is via the definition of the sequence, which told us that the ith term is HALF the (i - 1)th term. (In other words, each term is HALF the preceding term.)
Hence x4 = (x5)/2
We also have the algebraic equation relating x4 to x5 given in Statement 2: x5 = x4/(x4 + 1).
Combining the two, we know x5 = x5, so (x4)/2 = x4/(x4 + 1). Notice that the numerators are equal, so the denominators are also equal, and (x4 + 1) = 2. This tells us x4 = 1.
We can now work backwards from x4 using the definition of the sequence. Since each term is double the preceding term, x3 = 2, or double x4. Then x2 = 4 and x1 = 8 ... and we're done!
