\(A=\frac{2.2016}{1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+2016}}\)

\(A=\frac{2.2016}{1+\frac{1}{2.3:2}+\frac{1}{3.4:2}+\frac{1}{4.5:2}+..+\frac{1}{2016.2017:2}}\)

\(A=\frac{4032}{1+\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{2016.2017}}\)

\(A=\frac{4032}{1+2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+..+\frac{1}{2016.2017}\right)}\) .

\(A=\frac{4032}{1+2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2016}-\frac{1}{2017}\right)}\)

\(A=\frac{4032}{1+2\left(\frac{1}{2}-\frac{1}{2017}\right)}=\frac{4032}{1+\frac{2015}{2017}}\)

\(A=2017\)

Mẫu số \(=1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+2016}\)

\(=\frac{1}{\left(0+1\right).2:2}+\frac{1}{\left(1+2\right).2:2}+\frac{1}{\left(1+3\right).3:2}+\frac{1}{\left(1+4\right).4:2}+...+\frac{1}{\left(1+2016\right).2016:2}\)

\(=\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{2016.2017}\)

\(=2.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{2016.2017}\right)\)

\(=2.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2016}-\frac{1}{2017}\right)\)

\(=2.\left(1-\frac{1}{2017}\right)\)

\(=2.\frac{2016}{2017}=2.2016:2017\)

\(A=\left(2.2016\right):\left(2.2016:2017\right)\)

\(A=2.2016:2:2016.2017\)

\(A=2017\)

\(B=\frac{2,5-4.\left(\frac{5}{2}-1,2\right)+\frac{3}{8}}{4.\left(\frac{5}{2}-1,2\right)-\frac{3}{5}:\frac{2}{5}}-\frac{55}{148}\)

\(B=\frac{\frac{5}{2}-4.\left(\frac{25}{10}-\frac{12}{10}\right)+\frac{3}{8}}{4.\left(\frac{25}{10}-\frac{12}{10}\right)-\frac{3}{5}.\frac{5}{2}}-\frac{55}{148}\)

\(B=\frac{\frac{5}{2}-4.\frac{13}{10}+\frac{3}{8}}{4.\frac{13}{10}-\frac{3}{2}}-\frac{55}{148}\)

\(B=\frac{\frac{5}{2}-\frac{26}{5}+\frac{3}{8}}{\frac{26}{5}-\frac{3}{2}}-\frac{55}{148}\)

\(B=\frac{\frac{100}{40}-\frac{208}{40}+\frac{15}{40}}{\frac{52}{10}-\frac{15}{10}}-\frac{55}{148}\)

\(B=\frac{-\frac{93}{40}}{\frac{37}{10}}-\frac{55}{148}\)

\(B=\frac{93}{148}-\frac{55}{148}\)

\(B=\frac{19}{74}\)

\(\frac{2^{15}.9^4}{6^6.8^3}=\frac{2^{15}.\left(3^2\right)^4}{3^6.2^6.\left(2^3\right)^3}=\frac{2^{15}.3^8}{3^6.2^{15}}=\frac{3^8}{3^6}=3^2=9\)

Phân tích hết thừa số ra thừa số nguyên tố bạn ak. VD: 9^4 = 3^8 

\(\left(-2\right).\left(-1\frac{1}{2}\right).\left(-1\frac{1}{3}\right)...\left(-1\frac{1}{2013}\right)\)

\(=\frac{-2}{1}.\frac{-3}{2}.\frac{-4}{3}...\frac{-2014}{2013}\)

\(=-\left(\frac{2}{1}.\frac{3}{2}.\frac{4}{3}...\frac{2014}{2013}\right)\)(vì có lẻ thừa số âm)

\(=-2014\)

1) Ta có: \(-1+\left(8-4x\right)^2\ge-1\)

Dấu "=" xảy ra khi và chỉ khi (8 - 4x)² = 0 => 8 - 4x = 0 => 4x = 8 => x = 2

Vậy GTNN của -1 + (8 - 4x)² là -1 khi và chỉ khi x = 2

2) Ta có: \(5-\left(2+3x\right)^4\le5\)

Dấu ''='' xảy ra khi và chỉ khi (2 + 3x)⁴ = 0 => 2 + 3x = 0 => 3x = -2 => x = -2/3

Vậy GTLN của 5 - (2 + 3x)⁴ là 5 khi và chỉ khi x = -2/3

(8-4x)2 >=0 nên -1+(8-4x)² >=-1 nên GTNN: -1

Tương tự (2+3x)⁴ >=0 nên GTLN: 5

Ta có:

\(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\)

=> \(\frac{2x-2}{4}=\frac{3y-6}{9}=\frac{z-3}{4}\)

Áp dụng tính chất của dãy tỉ số = nhau ta có:

\(\frac{2x-2}{4}=\frac{3y-6}{9}=\frac{z-3}{16}=\frac{\left(2x-2\right)+\left(3y-6\right)-\left(z-3\right)}{4+9-4}\)

                                                      \(=\frac{2x+3y-z-5}{9}=\frac{50-5}{9}=\frac{45}{9}=5\)

=> \(\hept{\begin{cases}x-1=5.2=10\\y-2=5.3=15\\z-3=5.4=20\end{cases}}\)=> \(\hept{\begin{cases}x=11\\y=17\\z=23\end{cases}}\)

( x - 1,7 ) = 2,3

x = 2,3 + 1,7

x = 4

TH1

x-1,7=2,3 => x=4

TH2

x-1,7=-2,3 =>x =-0,6

\(A=\frac{1}{100}-\frac{1}{100.98}-\frac{1}{98.96}-....-\frac{1}{6.4}-\frac{1}{4.2}\)

\(\Rightarrow A=\frac{1}{100}-\left(\frac{1}{100.98}+\frac{1}{98.96}+....+\frac{1}{6.4}+\frac{1}{4.2}\right)\)

\(\Rightarrow A=\frac{1}{100}-\left(\frac{1}{100}-\frac{1}{98}+\frac{1}{98}-\frac{1}{96}+.....+\frac{1}{6}-\frac{1}{4}+\frac{1}{4}-\frac{1}{2}\right)\)

\(\Rightarrow A=\frac{1}{100}-\left(\frac{1}{100}-\frac{1}{2}\right)\Rightarrow A=\frac{1}{100}-\frac{1}{100}+\frac{1}{2}\Rightarrow A=\frac{1}{2}\)

\(A=\frac{1}{100}-\frac{1}{100.98}-\frac{1}{98.96}-...-\frac{1}{6.4}-\frac{1}{4.2}\)

\(A=\frac{1}{100}-\left(\frac{1}{2.4}+\frac{1}{4.6}+...+\frac{1}{96.98}+\frac{1}{98.100}\right)\)

\(A=\frac{1}{100}-\frac{1}{2.2}.\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{48.49}+\frac{1}{49.50}\right)\)

\(A=\frac{1}{100}-\frac{1}{4}.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{48}-\frac{1}{49}+\frac{1}{49}-\frac{1}{50}\right)\)

\(A=\frac{1}{100}-\frac{1}{4}.\left(1-\frac{1}{50}\right)\)

\(A=\frac{1}{100}-\frac{1}{4}.\frac{49}{50}\)

\(A=\frac{2}{200}-\frac{49}{200}=-\frac{47}{200}\)