Cho hai biểu thức A = \(\dfrac{1}{1.2} + \dfrac{1}{3.4} +...+ \dfrac{1}{49.50}\) và B = \(\dfrac{1}{26.50} + \dfrac{1}{27.49} +...+ \dfrac{1}{50.26} \)
Chứng minh rằng: A - 38B = 0
K
Khách
Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Những câu hỏi liên quan
18 tháng 9 2021
\(\dfrac{1}{1\cdot2}+\dfrac{1}{3\cdot4}+\dfrac{1}{5\cdot6}+...+\dfrac{1}{49\cdot50}\\ =1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}-\dfrac{1}{6}+...+\dfrac{1}{49}-\dfrac{1}{50}\\ =\left(1+\dfrac{1}{3}+\dfrac{1}{5}+...+\dfrac{1}{49}\right)-\left(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{6}+...+\dfrac{1}{50}\right)\\ =\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{50}\right)-2\left(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{6}+...+\dfrac{1}{50}\right)\\ =\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{50}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{25}\right)\)
\(=\dfrac{1}{26}+\dfrac{1}{27}+...+\dfrac{1}{49}+\dfrac{1}{50}\)
12 tháng 7 2017
2, a-b=ab => a=ab+b => a=b(a+1)
thay a=b(a+1) vào a:b ta có: => b:b(a+1)=a+1
Theo bài ra ta có: a:b=a-b
=> a+1=a-b
=>-b=1
=> b=-1
Thay b=-1 vào a-b=ab ta có : a-(-1)=-a
=> a +1=-a
=>a=-1/2
Vậy a=-1/2. b=-1
26 tháng 7 2017
a, \(\dfrac{1}{2!}+\dfrac{2}{3!}+...+\dfrac{99}{100!}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{99.100}\)
\(=\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100}=1-\dfrac{1}{100}< 1\)
\(\Rightarrowđpcm\)
d, \(D=\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{99}}\)
\(\Rightarrow3D=1+\dfrac{1}{3}+...+\dfrac{1}{3^{98}}\)
\(\Rightarrow3D-D=\left(1+\dfrac{1}{3}+...+\dfrac{1}{3^{98}}\right)-\left(\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{99}}\right)\)
\(\Rightarrow2D=1-\dfrac{1}{3^{99}}\)
\(\Rightarrow D=\dfrac{1}{2}-\dfrac{1}{3^{99}.2}< \dfrac{1}{2}\)
\(\Rightarrowđpcm\)
26 tháng 7 2017
\(\dfrac{1}{1.2}+\dfrac{1}{3.4}+...+\dfrac{1}{49.50}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{49}-\dfrac{1}{50}\)
\(=\left(1+\dfrac{1}{3}+...+\dfrac{1}{49}\right)-\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{50}\right)\)
\(=\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{49}+\dfrac{1}{50}\right)-2\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{50}\right)\)
\(=1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{49}+\dfrac{1}{50}-1-\dfrac{1}{2}-...-\dfrac{1}{25}\)
\(=\dfrac{1}{26}+\dfrac{1}{27}+...+\dfrac{1}{50}\)
\(\Rightarrowđpcm\)
10 tháng 6 2023
Ta biến đổi \(A=\dfrac{2-1}{1.2}+\dfrac{4-3}{3.4}+...+\dfrac{2016-2015}{2016.2015}+\dfrac{2018-2017}{2017.2018}\)
\(A=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2015}-\dfrac{1}{2016}+\dfrac{1}{2017}-\dfrac{1}{2018}\)
\(A=\left(1+\dfrac{1}{3}+...+\dfrac{1}{2017}\right)-\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2018}\right)\)
\(A=\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2017}\right)-2\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2018}\right)\)
\(A=\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2017}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{1009}\right)\)
\(A=\dfrac{1}{1010}+\dfrac{1}{1011}+...+\dfrac{1}{2017}+\dfrac{1}{2018}\)
Lại có \(B=\dfrac{1}{1010.2018}+\dfrac{1}{1011.2017}+...+\dfrac{1}{2018.1010}\)
\(B=\dfrac{1}{3028}.\left(\dfrac{3028}{1010.2018}+\dfrac{3028}{1011.2017}+...+\dfrac{3028}{2018.1010}\right)\)
\(B=\dfrac{1}{3028}\left(\dfrac{1}{1010}+\dfrac{1}{2018}+\dfrac{1}{1011}+\dfrac{1}{2017}+...+\dfrac{1}{2018}+\dfrac{1}{1010}\right)\)
\(B=\dfrac{1}{3028}.2\left(\dfrac{1}{1010}+\dfrac{1}{1011}+...+\dfrac{1}{2018}\right)\)
\(B=\dfrac{1}{3028}.2A\) \(\Rightarrow\dfrac{A}{B}=1514\inℤ\). Ta có đpcm
11 tháng 9 2023
\(A=\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{49.50}\)
\(A=\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{49}-\dfrac{1}{50}\)
\(A=\dfrac{1}{1}-\dfrac{1}{50}\)
\(A=\dfrac{49}{50}\)
29 tháng 4 2017
a) Để A là phân số thì 5 không chia hết cho n-1 hay n-1 không phải Ư(5) mà Ư(5)={-5;-1;1;5}
Ta có bảng sau:
| \(n-1\ne\) | -5 | -1 | 1 | 5 |
| \(n\ne\) | -4 | 0 | 2 | 6 |
Vậy n\(\ne\left\{-4;0;2;6\right\}\)thì A là phân số
n=0 => A=\(\dfrac{5}{0-1}=-5\)
n=10 => A=\(\dfrac{5}{10-1}=\dfrac{5}{9}\)
n=-2 => A=\(\dfrac{5}{-2-1}=-\dfrac{5}{3}\)
Để A là số nguyên =>5 chia hết cho n-1 <=>n-1 là Ư(5)
Từ bảng trên => n={-4;0;2;6} thì A nguyên
b) Do n là Số tự nhiên => n và n+1 là 2 số tự nhiên liên tiếp
=>n và n+1 nguyên tố cùng nhau
=>phân số \(\dfrac{n}{n+1}\)tối giản(dpcm)
c)\(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{49\cdot50}=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{49}-\dfrac{1}{50}=1-\dfrac{1}{50}< 1\left(đpcm\right)\)
29 tháng 4 2017
c) 1/1.2 + 1/2.3 + 1/3.4 + .....+ 1/49.50
= 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + ......+ 1/49 - 1/50
tới bước đây mik làm gọn lại chút nha
= 1/1 - 1/50
=49/50
Suy ra : 49/50 <1 ( điều phải chứng minh )
Dạng hay :v
Ta có:
\(A = \dfrac{1}{1.2} + \dfrac{1}{3.4} +...+ \dfrac{1}{49.50}\)
\(=>A=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{49}-\dfrac{1}{50}\)
\(=>A=(1+\dfrac{1}{3}+...+\dfrac{1}{49})-(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{50})\)
\(=>A=(1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{49}+\dfrac{1}{50})-2.(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{50})\)
\(=>A=(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{50})-(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{25})\)
\(=>A=\dfrac{1}{26}+\dfrac{1}{27}+...+\dfrac{1}{50} (1)\)
Ta lại có:
\(B = \dfrac{1}{26.50} + \dfrac{1}{27.49} +...+ \dfrac{1}{50.26}\)
\(=>38B=\dfrac{38}{26.50}+\dfrac{38}{27.49}+...+\dfrac{38}{50.26}\)
\(=>38B=\dfrac{76}{26.50}+\dfrac{76}{27.49}+...+\dfrac{38}{38.38}\)
\(=>38B=\dfrac{1}{26}+\dfrac{1}{50}+\dfrac{1}{27}+\dfrac{1}{49}+...+\dfrac{1}{38}\)
\(=>38B=\dfrac{1}{26}+\dfrac{1}{27}+...+\dfrac{1}{50} (2)\)
Từ (1)(2):
\(=>A = 38B\)
\(=>A-38B=0\)