## Sum of Terms

In the sequence of numbers $2,5,3,\cdots$, each term (starting with the third) is equal to the term preceding it minus the term preceding that one. What is the sum of the first $100$ terms of this sequence?
Source: NCTM Mathematics Teacher, December 2005

SOLUTION
The terms are $2,5,3,\textrm{-}2,\textrm{-}5,\textrm{-}3,2,5,3,\textrm{-}2,\textrm{-}5,\textrm{-}3,2,5,3,\textrm{-}2,\textrm{-}5,\textrm{-}3,\cdots$
The sum of every six terms equal $0$. If we divide the first $100$ terms into six, we get
$100=16(6)+4$
All the first $96$ terms sum to zero and the last four terms sum to
$2+5+3+(\textrm{-}2)=8$

Answer: $8$