Evaluate the alternating triple pattern: 1 − 2 − 3 + 2 − 3 − 4 + 3 − 4 − 5 + … up to 100 terms. Find the total sum.

Introduction / Context:
This series groups naturally into blocks of three terms: +n, −(n+1), −(n+2). Summing by blocks streamlines evaluation of long finite series without writing every term.

Given Data / Assumptions:

• Patterned blocks: for n = 1, 2, 3, … each block is n − (n+1) − (n+2).
• Total terms required: 100.

Concept / Approach:
First compute the sum per block S_n, then determine how many full 3-term blocks fit in 100, and add any leftover initial terms of the next block.

Step-by-Step Solution:
Block sum: S_n = n − (n+1) − (n+2) = −(n+3).In 100 terms there are 33 full blocks (99 terms) and 1 extra term.Sum of 33 blocks = −∑_{n=1}^{33} (n+3) = −( (33*34)/2 + 3*33 ) = −(561 + 99) = −660.The 100th term is the first term of the 34th block, i.e., +34.Total = −660 + 34 = −626.

Verification / Alternative check:
Spot-check early partial sums (after 3, 6, 9 terms) to confirm the −(n+3) block behavior before trusting the general calculation.

Why Other Options Are Wrong:
−624 and −676 reflect arithmetic slips (e.g., miscounting blocks or the final leftover term). −694 is too negative.

Common Pitfalls:
Forgetting the lone 100th term after 33 blocks; sign errors when combining positives and negatives in the block.

Final Answer:
-626