Question

[tớ thích cậu]

1/2+5/6+11/12+..................379/380=a/b là phân số tối giản a+b =

Answer 1

[subjects]

\(\dfrac{1}{2}+\dfrac{5.}{6}+\dfrac{11}{12}+...+\dfrac{379}{380}\\ =\left(1-\dfrac{1}{2}\right)+\left(1-\dfrac{1}{6}\right)+\left(1-\dfrac{1}{12}\right)+...+\left(1-\dfrac{1}{380}\right)\\ =\left(1+1+1+.....+1\right)-\left(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{380}\right)\\ =\left(1+1+1+.....+1\right)-\left(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{19\cdot20}\right)\\ =19-\left(\dfrac{1}{1}-\dfrac{1}{20}\right)\\ =19-\dfrac{1}{1}+\dfrac{1}{20}=\dfrac{361}{20}=\dfrac{a}{b}\)

vậy a + b = 361 + 20 = 381

Answer 2

[Nguyễn Đức Trí]

\(S=\frac12+\frac56+\frac{11}{12}+\cdots+\frac{379}{380}\)

\(S=\frac{1.2-1}{1.2}+\frac{2.3-1}{2.3}+\frac{3.4-1}{3.4}+\cdots+\frac{19.20-1}{19.20}\)

\(S=\frac12+\left(1-\frac12+\frac13\right)+\left(1-\frac13+\frac14\right)+\cdots+\left(1-\frac{1}{19}+\frac{1}{20}\right)\)

\(S=18+\frac{1}{20}=\frac{361}{20}=\frac{a}{b}\)

\(\rArr a=361;b=20\)

\(\rArr a+b=361+20=381\)