Suppose that you are given a sorted sequence of distinct integers {a1,a2,…,an}. give an o(lgn) algorithm to determine whether there exists an i index such as ai

Since the sequence is sorted, you can use the binary search, which is of O(lg2).  You can google binary search, or inspire from the following.
Basically, the algorithm looks like the following:

checkMiddle(ai, a1,n)
if (n==0) return(-1)   // key ai not found
k=(1+n)/2;
if (a_k = ai)  return(k)
else
if (a_k>ai
   checkMiddle(ai,a_1,k)
else
  checkMiddle(ai,a_(k+1),n-k)
endif

You may have to tune and check, but the basic idea is there.

Also, concerning computing algorithms, you are better off with posting it under technology and computers.

55, 90 and 35 (when you add it, it should be 180)