Using the relationship [A=1, B=2,..... Z=26] provide a word for each tier of the triangle such that:
1. The letter sum per tier is the largest triangle possible.
2. The letter sum per tier is the smallest triangle possible.

While TIG = 36 is a triangle it is not appropriate for tier three in either of the tasks #1 or #2 above.

Are you able to extend either triangle beyond 6 tiers?

(In reply to computer solution by Charlie)

Taxonomy of Wordplay is a site with various word trivia, including a list of shortest and longest words with smallest and largest "letter sum" values (though not for the specific nature of triangular values).
From the 'smaller value' list are words that have a smaller triangular value than in Charlie's list, and probably due to these words are listed as uncommon:

Word   Word in         Word        Word in             Word
Length Charlie's List  Value Tri  ToW                  Value Tri
6      baccae               15     5   bacaba                 10    4 (-1)  
13    beneficiaries       105   14  anacardiaceae       66  11 (-3)  
16    accidentalnesses 153   17  lactobacillaceae    105 14 (-3)

From the 'larger value' list is an uncommon word that has a larger triangular value:

Word    Word in         Word       Word in                Word
Length  Charlie's List  Value Tri  ToW                    Value Tri
16    counterterrorism 231   21  unpresumptuously 276   23 (+2)
            

Posted by Dej Mar on 2012-09-08 12:36:10