If we take every other term, starting with the second term 2, we get \[2, 5, 11, 23, 47, \dots.\]If we add one to each of these terms, we get \[3, 6, 12, 24, 48, \dots.\]Each term appear to be double the previous term. To confirm this, let one term in the original sequence be $ x - 1, $ after we have added 1. Then the next term is $ 2(x - 1) = 2x - 2, $ and the next term after that is $ 2x - 2 + 1 = 2x - 1 $. This confirms that in the sequence 3, 6, 12, 24, 48, $ \dots, $ each term is double the previous term. Then the 50th term in this geometric sequence is $ 3 \cdot 2^{49}, $ so the 100th term in the original sequence is $ 3 \cdot 2^{49} - 1, $ so $ k = \boxed{49} $.