There is a straight line which keep getting new "ends" and creating new segments. First, it has 2 then 3 ends,4, 5 6 etc. As you advance adding new ends, the number of segments icreases in a larger number than the number of ends. The first 2 is only 1 segment, when it's 3 is 3 segments, when it's 4 it's 6 segments, with 5 it's 10 segments, etc. I figure that the first term of the conjecture is n-1 when "n" is the number of ends. For instance: for 2 ends: n-1 = 1 For 3 ends: (n-1) + 1 = 2 +1 = 3 for 4 ends: (n-1) + 2 + 1 = 6 for 5 ends: (n-1) + 3 + 2 + 1 = 10 and so on and so forth. Every time you increase the number of ends by one, you need to add the new number (n-1) plus the previous number of segments. The question is to design the abbreviated form to solve just by giving the number of ends. A second question is add all the "odd" number of ends. In this formula. Thank you!