Dễ thấy M là dãy Fibonacci
\(M=1+2+3+5+8+13+21+34+55+89+144+233\\ M=3+3+13+13+55+55+233+233\\ M=2\times\left(3+13+55+233\right)=2\times304=608\)
\(N=\dfrac{2+3}{2\times3}-\dfrac{3+4}{3\times4}+...-\dfrac{25+26}{25\times26}+\dfrac{26+27}{26\times27}\\ N=\dfrac{1}{3}+\dfrac{1}{2}-\dfrac{1}{4}-\dfrac{1}{3}+....-\dfrac{1}{26}-\dfrac{1}{25}+\dfrac{1}{27}+\dfrac{1}{26}\\ N=\dfrac{1}{2}+\dfrac{1}{27}=\dfrac{29}{54}\)
\(K=\dfrac{38}{25}+\left(\dfrac{9}{10}-\dfrac{11}{15}\right)+\left(\dfrac{13}{21}-\dfrac{15}{28}\right)+\left(\dfrac{17}{36}-\dfrac{19}{45}\right)+...+\left(\dfrac{197}{4851}-\dfrac{199}{4950}\right)+\dfrac{53}{26\times27}\\ K=\dfrac{38}{25}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+...+\dfrac{1}{2450}+\dfrac{53}{26\times27}\\ K=\dfrac{38}{25}+\dfrac{1}{2\times3}+\dfrac{1}{3\times4}+...+\dfrac{1}{49\times50}+\dfrac{53}{26\times27}\\ K=\dfrac{38}{25}+\dfrac{53}{26\times27}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{49}-\dfrac{1}{50}\\ K=\dfrac{38}{25}+\dfrac{53}{26\times27}+\dfrac{1}{2}-\dfrac{1}{50}=\dfrac{3}{2}+\dfrac{1}{2}+\dfrac{53}{702}=1+\dfrac{53}{702}=\dfrac{755}{702}\)
4 và 5
