Final answer:
The problem is a telescoping series where the term (-2)/((n+1)/(n+2)) simplifies and the series can be evaluated by realizing that terms cancel out, leaving only the initial and final terms after simplification.
Step-by-step explanation:
The telescoping series method is a technique used to evaluate certain types of series where each term cancels out a part of the previous or next term. In this case, we want to evaluate the finite sum of (-2)/((n+1)/(n+2)) which when simplified, can actually take the form of any of the provided options, a), b), c), or d). The most performed option for telescoping would be option c) ((1)/(n+1) - (1)/(n+2)) as adding sequential terms would start cancelling out the intermediate terms, leaving only the first and last terms of the series after simplification.
To proceed with the telescoping, we simplify the original expression:
(-2) ÷ ((n+1)/(n+2)) = (-2)(n+2)/(n+1) = -2 - (4/(n+1)) + (4/(n+2))
When we add these up over a series from n=1 to n=k, we notice that all the terms in the middle will cancel out, leaving only terms from the first and the last part of the summation.