N = 1 - 2/2.3 + 1 - 2/3.4 +.....+ 1 - 2/99.100

   = 98 - 2.(1/2.3 + 1/3.4 + ...... + 1/99.100)

   = 98 - 2.(1/2-1/3+1/3-1/4+....+1/99-1/100)

   = 98 - 2.(1/2-1/100)

   = 98 - 2.49/100 = 98-49/50 < 98

Mà 49/50 < 1

=> N > 98-1 = 97

=> 97 < N < 98

Tk mk nha

Ta có công thức tổng quát của số hạng trong tổng trên có dạng:

\(x_n=\frac{n\left(n+3\right)}{\left(n+1\right)\left(n+2\right)}=\frac{n^2+3n+2-2}{n^2+3n+2}\)

\(=1-\frac{2}{n^2+3n+2}=1-\frac{2}{\left(n+1\right)\left(n+2\right)}\)

\(\Rightarrow\frac{1.4}{2.3}=1-\frac{2}{2.3}\)

\(\frac{2.5}{3.4}=1-\frac{2}{3.4}\)

\(\frac{3.6}{4.5}=1-\frac{2}{4.5}\)

....

\(\frac{98.101}{99.100}=1-\frac{2}{99.100}\)

\(\Rightarrow N=98-2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}\right)\)

\(=98-2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\right)\)

\(=98-2\left(\frac{1}{2}-\frac{1}{100}\right)\)

\(=98-1+\frac{1}{50}=97+\frac{1}{50}\)

Vậy 97 < N < 98

Ta có \(\frac{a\left(a+3\right)}{\left(a+1\right)\left(a+2\right)}=\frac{\left(a+1-1\right)\left(a+2+1\right)}{\left(a+1\right)\left(a+2\right)}=\frac{\left(a+1\right)\left(a+2\right)-\left(a+2\right)+\left(a+1\right)-1}{\left(a+1\right)\left(a+2\right)}\\ \)

= \(1-\frac{2}{\left(a+1\right)\left(a+2\right)}\)

Áp dụng ta có N = \(98-\left(\frac{2}{2.3}+...+\frac{2}{99.100}\right)=98-2.\left(\frac{1}{2.3}+...+\frac{1}{99.100}\right)=98-2.\left(\frac{1}{2}-\frac{1}{100}\right)>97\)

Bạn tham khảo link này: https://h.vn/hoi-dap/question/537598.html

E=\(\frac{1.2.3.....97.98}{2.3.4.....98.99}\)+\(\frac{4.5.6....100.101}{3.4.5...99.100}\)

E=\(\frac{1}{99}\)+\(\frac{101}{3}\)

E=\(\frac{304}{99}\)