deletions | additions       diff --git a/Version3.tex b/Version3.tex  index 7dc3a73..6b8a6e3 100644  --- a/Version3.tex  +++ b/Version3.tex 
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$$T(n) = a + \sum_{i=0}^{n-1} f_3(n) = a + \sum_{i=0}^{n-1} (b + \sum_{j=i}^{n-1} (c + \sum_{k=i}^{j} d))$$  Each loop in the method corresponds to one sum or one method $f_i$ where the lowest index is the innermost sum/loop.  We define the other functions:     $$f_1(j,i) = \sum_{k=i}^j d = d \sum_{k=i}^j 1 = d(1 + j - i) = d + dj - di$$  $$  f_1(j,i) f_2(n,i)  = \sum_{k=i}^j d \sum_{j=i}^{n-1} (c + f_1(j,i))  = d \sum_{k=i}^j c\sum_{j=i}^{n-1}  1= d(1  + j - i) = d d\sum_{j=i}^{n-1} 1  + dj d\sum_{j=i}^{n-1} j  - di di\sum_{j=i}^{n-1} 1  $$