Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Ta có 1.4/2.3=(2-1)(3+1)/2.3=1-1/2+1/3-1/2.3
2.5/3.4=(3-1)(4+1)/3.4=1-1/3+1/4-1/3.4
...
Suy ra N=(1-1/2+1/3-1/2.3)+(1-1/3+1/4-1/3.4)+....+(1-1/99+1/100-1/99.100)
N=\(98+\dfrac{1}{100}-\dfrac{1}{2}-\dfrac{1}{2.3}-\dfrac{1}{3.4}-....-\dfrac{1}{99.100}\)
Xét P=\(\dfrac{1}{2.3}+\dfrac{1}{3.4}+....+\dfrac{1}{99.100}\)
P=\(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+.....+\dfrac{1}{99}-\dfrac{1}{100}\)
P=\(\dfrac{1}{2}-\dfrac{1}{100}\)
Vậy N=98-1+\(\dfrac{1}{50}\)
N=\(97+\dfrac{1}{50}\)
Vậy 97<N<98(ĐPCM)
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 = \(98-\left(\frac{2}{2.3}+...+\frac{2}{99.100}\right)=98-2.\left(\frac{1}{2.3}+...+\frac{1}{99.100}\right)=98-2.\left(\frac{1}{2}-\frac{1}{100}\right)>97\)
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=\(\frac{304}{99}\)
\(A=\frac{4}{6}+\frac{10}{12}+\frac{18}{20}+...+\frac{9898}{9900}\)
\(A=1-\frac{2}{6}+1-\frac{2}{12}+1-\frac{2}{20}+...+1-\frac{2}{9900}\)
\(A=98-\left(\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{99.100}\right)\)Đặt Biểu thức trong ngoặc đơn là B
\(\Rightarrow A=98-B\)
\(\Rightarrow\frac{B}{2}=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}\)
\(\frac{B}{2}=\frac{3-2}{2.3}+\frac{4-3}{3.4}+\frac{5-4}{4.5}+...+\frac{100-99}{99.100}\)
\(\frac{B}{2}=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}=\frac{1}{2}-\frac{1}{100}=\frac{49}{100}\)
\(\Rightarrow B=\frac{2.49}{100}=\frac{98}{100}\)
Ta nhận thấy \(B=\frac{98}{100}< 1\Rightarrow A=98-\frac{98}{100}=97+\frac{2}{100}\)
\(\Rightarrow97< A< 98\left(dpcm\right)\)
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