So sánh :
M = 1 + ( 1 + 2 ) + ( 1 + 2 + 3 ) + ....+ ( 1 + 2 + 3 + ......+ 99 )
N = 1. 99 + 2 . 98 + 3 . 97 + ....... + 99 . 1
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\(M=\frac{99}{1}+\frac{98}{2}+\frac{97}{3}+...+\frac{2}{98}+\frac{1}{99}\)
cộng vào mỗi phân số trong 98 phân số sau,trừ phân số cuối đi 98 , ta được :
\(M=1+\left(\frac{98}{2}+1\right)+\left(\frac{97}{3}+1\right)+...+\left(\frac{2}{98}+1\right)+\left(\frac{1}{99}+1\right)\)
\(M=\frac{100}{100}+\frac{100}{2}+\frac{100}{3}+...+\frac{100}{98}+\frac{100}{99}\)
chuyển phân số \(\frac{100}{100}\)ra sau , ta được :
\(M=\frac{100}{2}+\frac{100}{3}+...+\frac{100}{98}+\frac{100}{99}+\frac{100}{100}\)
\(M=100.\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{98}+\frac{1}{99}+\frac{1}{100}\right)\)
\(\Rightarrow\frac{M}{N}=\frac{100.\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{98}+\frac{1}{99}+\frac{1}{100}\right)}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{99}+\frac{1}{100}}=100\)
\(\frac{T}{M}=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}}{\frac{1}{99}+\frac{2}{98}+...+\frac{98}{2}+\frac{99}{1}}\)
Xét M - 99 + 98 = \(\frac{100}{99}+\frac{100}{98}+...+\frac{100}{2}\)
\(\Leftrightarrow M-1=100\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}\right)\)
\(\Rightarrow M=\frac{100}{100}+100\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}\right)=100\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{100}\right)\)
\(\Rightarrow\frac{T}{M}=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}}{100\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{100}\right)}=\frac{1}{100}\)
Ta có: \(M=\dfrac{\dfrac{1}{99}+\dfrac{2}{98}+\dfrac{3}{97}+\dfrac{4}{96}+...+\dfrac{97}{3}+\dfrac{98}{2}+\dfrac{99}{1}}{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+...+\dfrac{1}{100}}\)
\(=\dfrac{\left(1+\dfrac{1}{99}\right)+\left(1+\dfrac{2}{98}\right)+\left(1+\dfrac{3}{97}\right)+\left(1+\dfrac{4}{96}\right)+...+\left(1+\dfrac{98}{2}\right)+1}{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+...+\dfrac{1}{100}}\)
\(=\dfrac{\dfrac{100}{99}+\dfrac{100}{98}+\dfrac{100}{97}+...+\dfrac{100}{1}+\dfrac{100}{2}}{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+...+\dfrac{1}{100}}\)
=100
Ta có: \(N=\dfrac{92-\dfrac{1}{9}-\dfrac{2}{10}-\dfrac{3}{11}-...-\dfrac{90}{98}-\dfrac{91}{99}-\dfrac{92}{100}}{\dfrac{1}{45}+\dfrac{1}{50}+\dfrac{1}{55}+...+\dfrac{1}{495}+\dfrac{1}{500}}\)
\(=\dfrac{\left(1-\dfrac{1}{9}\right)+\left(1-\dfrac{2}{10}\right)+\left(1-\dfrac{3}{11}\right)+...+\left(1-\dfrac{90}{98}\right)+\left(1-\dfrac{91}{99}\right)+\left(1-\dfrac{92}{100}\right)}{\dfrac{1}{5}\left(\dfrac{1}{9}+\dfrac{1}{10}+\dfrac{1}{11}+...+\dfrac{1}{99}+\dfrac{1}{100}\right)}\)
\(=\dfrac{\dfrac{8}{9}+\dfrac{8}{10}+\dfrac{8}{11}+...+\dfrac{8}{99}+\dfrac{8}{100}}{\dfrac{1}{5}\left(\dfrac{1}{9}+\dfrac{1}{10}+\dfrac{1}{11}+...+\dfrac{1}{99}+\dfrac{1}{100}\right)}\)
\(=\dfrac{8}{\dfrac{1}{5}}=40\)
\(\Leftrightarrow\dfrac{M}{N}=\dfrac{100}{40}=\dfrac{5}{2}\)
M = 1 + ( 1 + 2 ) + ( 1 + 2 + 3 ) + ....+ ( 1 + 2 + 3 + ......+ 99 )
M gồm 99 tổng, số 1 có mặt ở 99 tổng, số 2 có mặt ở 98 tổng,......., số 98 có mặt ở 2 tổng, số 99 có mặt ở 1 tổng
Vậy:
M = 1.99 + 2.98 + ...... + 98.2 + 99.1 = N
Vậy M = N
Ta có:
M=1 + ( 1 + 2 ) + ( 1 + 2 + 3 ) + ....+ ( 1 + 2 + 3 + ......+ 99 )
=1+1+2+1+2+3+...+1+2+3+...+99
=(1+1+...+1+1)+(2+2+2+...+2)+...+(98+98)+99
-----99 số 1--; --98 số 2--------;...
=1.99+2.98+...+98.2+99.1
Mà N = 1. 99 + 2 . 98 + 3 . 97 + ....... + 99 . 1
=>M=N