Question

[Trần Võ Mỵ Nương]

Tính nhanh 

Nhìn thì tưởng dễ nhưng làm mới biết thực sự rất khó mn ak! 

Answer 1

[Xyz OLM]

\(A=1+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+...+8}\)

= \(\frac{1}{\frac{1.2}{2}}+\frac{1}{\frac{2.3}{2}}+\frac{1}{\frac{3.4}{2}}+...+\frac{1}{\frac{8.9}{2}}\)

= \(\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{8.9}=2\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{8.9}\right)\)

\(=2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{8}-\frac{1}{9}\right)=2\left(1-\frac{1}{9}\right)=2.\frac{8}{9}=\frac{16}{9}\)

Answer 2

[ŇɠʋĦ [ 𝕿𝕹ʊōɴɢ ( ɻɛɑm ʙáo cáo ) ]]

\(A=1+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+...+8}\)

\(A=\frac{1}{\frac{1.2}{2}}+\frac{1}{\frac{2.3}{2}}+\frac{1}{\frac{3.4}{2}}+...+\frac{1}{\frac{8.9}{2}}\)

\(A=\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{8.9}=2\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{8.9}\right)\)

\(A=2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{8}-\frac{1}{9}\right)\)

\(A=2\left(1-\frac{1}{9}\right)=2.\frac{8}{9}=\frac{16}{9}\)

\(\text{Vậy A }=\frac{16}{9}\)

\(\text{#Hok tốt!}\)