Question

Answer 1

\(A=\dfrac{-1}{199}-\dfrac{1}{199\cdot198}-\dfrac{1}{198\cdot197}-...-\dfrac{1}{3\cdot2}-\dfrac{1}{2}\)

\(A=\dfrac{-1}{199}-\left(\dfrac{1}{198}-\dfrac{1}{199}\right)-\left(\dfrac{1}{197}-\dfrac{1}{198}\right)-...-\left(\dfrac{1}{2}-\dfrac{1}{3}\right)-\dfrac{1}{2}\)

\(A=\dfrac{-1}{199}-\dfrac{1}{198}+\dfrac{1}{199}-\dfrac{1}{197}+\dfrac{1}{198}-...-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{2}\)

\(A=-\dfrac{1}{2}-\dfrac{1}{2}\)

\(A=\dfrac{-1-1}{2}\)

\(A=\dfrac{-2}{2}\)

\(A=-1\)

Answer 2

\(=\dfrac{-1}{199}-\left(\dfrac{1}{2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{198\cdot199}\right)\)

\(=\dfrac{-1}{199}-\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{198}-\dfrac{1}{199}\right)\)

=-1/199-(1-1/199)

=-1/199-1+1/199=-1

Answer 3

\(A=\dfrac{-1}{199}-\dfrac{1}{199.198}-\dfrac{1}{198.197}-\dfrac{1}{197.196}-...-\dfrac{1}{3.2}-\dfrac{1}{2}\)

\(=-\dfrac{1}{199}-\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{197.198}+\dfrac{1}{198.199}\right)\)

\(=-\dfrac{1}{199}-\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-...-\dfrac{1}{197}+\dfrac{1}{197}-\dfrac{1}{198}+\dfrac{1}{198}-\dfrac{1}{199}\right)\)

\(=-\dfrac{1}{199}-\left(1-\dfrac{1}{199}\right)\)

\(=-\dfrac{1}{199}-\dfrac{198}{199}=-\dfrac{199}{199}=-1\)