a)A=\(\frac{1}{100}-\frac{1}{100.99}-\frac{1}{99.98}-\frac{1}{98.97}-...-\frac{1}{3.2}-\frac{1}{2.1}\)

      =\(\frac{1}{100}-\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{97.98}+\frac{1}{98.99}+\frac{1}{99.100}\right)\)

      =\(\frac{1}{100}-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{97}-\frac{1}{98}+\frac{1}{98}-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\right)\)

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

      =\(\frac{1}{100}-\frac{99}{100}\)

      =\(\frac{-98}{100}=\frac{-49}{50}\)

a) $A= \backslash (\backslash frac\{1\}\{100\}-\backslash frac\{1\}\{100.99\}-\backslash frac\{1\}\{99.98\}-...-\backslash frac\{1\}\{3.2\}-\backslash frac\{1\}\{2.1\}\backslash )$ 

$A=$$\backslash (\backslash frac\{1\}\{100\}-\backslash left(\backslash frac\{1\}\{1.2\}+\backslash frac\{1\}\{2.3\}+...+\backslash frac\{1\}\{98.99\}+\backslash frac\{1\}\{99.100\}\backslash right)\backslash )$$\mathrm{(\; vận\; dụng quy tắc\; dấu\; ngoặc)}$\(A=\frac{1}{100}-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{98}-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\right)\)

$\mathrm{\backslash (A=\backslash frac\{1\}\{100\}-\backslash left(1-\backslash frac\{1\}\{100\}\backslash right)\backslash ) = \backslash (\backslash frac\{1\}\{100\}-\backslash frac\{99\}\{100\}\backslash )}$ = \(\frac{-98}{100}\) = \(\frac{-49}{50}\)

$\frac{\mathrm{A=1/}}{100}-\frac{\mathrm{1/}}{100.99}-\frac{\mathrm{1/}}{\mathrm{99.98\; -...}}-\frac{\mathrm{1/}}{3.2}-\frac{\mathrm{1/}}{2.1}$

$A=\frac{\mathrm{1/}}{100}-(\frac{\mathrm{1/}}{1.2}+\frac{\mathrm{1/}}{2.3}+...+\frac{\mathrm{1/}}{98.99}+\frac{\mathrm{1/}}{99.100})$

 (vận dụng quy tắc dấu ngoặc)
$A=\frac{\mathrm{1/}}{100}-(1-\frac{\mathrm{1/}}{2}+\frac{\mathrm{1/}}{2}-\frac{\mathrm{1/}}{3}+...+\frac{\mathrm{1/}}{98}-\frac{\mathrm{1/}}{99}+\frac{\mathrm{1/}}{99}-$

$\frac{\mathrm{1/}}{\mathrm{100\; )}}$

 
$A=\frac{\mathrm{1/}}{100}-(1-\frac{\mathrm{1/}}{\mathrm{100)}}$ ) = $\frac{\mathrm{1/}}{100}-\frac{\mathrm{99/}}{100}=\frac{-\mathrm{98/}}{100}=\frac{-\mathrm{49/}}{50}$

/ là phần nha

Giải:

a) \(\left(9\dfrac{4}{9}+5\dfrac{2}{3}\right)-5\dfrac{1}{2}\) 

\(=\left(\dfrac{85}{9}+\dfrac{17}{3}\right)-\dfrac{11}{2}\) 

\(=\dfrac{136}{9}-\dfrac{11}{2}\) 

\(=\dfrac{173}{18}\) 

b) \(\dfrac{13}{9}.\dfrac{15}{4}-\dfrac{13}{9}.\dfrac{7}{4}-\dfrac{13}{9}.\dfrac{5}{4}\) 

\(=\dfrac{13}{9}.\left(\dfrac{15}{4}-\dfrac{7}{4}-\dfrac{5}{4}\right)\) 

\(=\dfrac{13}{9}.\dfrac{3}{4}\) 

\(=\dfrac{13}{12}\) 

c) \(\dfrac{2}{3}+\dfrac{5}{8}-\dfrac{-1}{3}+0,375\) 

\(=\left(\dfrac{2}{3}-\dfrac{-1}{3}\right)+\left(\dfrac{5}{8}+\dfrac{3}{8}\right)\) 

\(=1+1\) 

\(=2\)

d) \(75\%-3\dfrac{1}{2}+1,5:\dfrac{10}{7}\) 

\(=\dfrac{3}{4}+\dfrac{7}{2}+\dfrac{3}{2}:\dfrac{10}{7}\) 

\(=\dfrac{3}{4}+\dfrac{7}{2}+\dfrac{21}{20}\) 

\(=\dfrac{53}{10}\) 

e) \(1\dfrac{13}{15}.\left(0,5\right)^2.3+\left(\dfrac{8}{15}-1\dfrac{19}{60}\right):1\dfrac{23}{24}\) 

\(=\dfrac{28}{15}.\dfrac{1}{4}.3+\left(\dfrac{8}{15}-\dfrac{79}{60}\right):\dfrac{47}{24}\) 

\(=\dfrac{7}{5}+\dfrac{-47}{60}:\dfrac{47}{24}\) 

\(=\dfrac{7}{5}+\dfrac{-2}{5}\) 

\(=1\)

\(S=\left(0,25\right)^2+\left(0,5\right)^2+...+\left(2,5\right)^2\)

\(\Rightarrow\frac{n\left(n+1\right)\left(2n+1\right)}{6}=\frac{2,5\left(2,5+1\right)\left(2,5.2+1\right)}{6}\)

\(\Rightarrow S=8,75\)

Ta có :
\(S=\left(0,25\right)^2+\left(0,5\right)^2+...+\left(2,5\right)^2\)

\(\Rightarrow4S=2^2.\left(0,25\right)^2+2^2.\left(0,5\right)^2+.....+2^2.\left(2,5\right)^2\)

\(\Rightarrow4S=1^2+2^2+....+10^2\)

\(\Rightarrow4S=385\)

\(\Rightarrow S=\frac{385}{4}\)

a)1,5.(1/3-2/3)                         

=3/2.(-1/3)

=-1/2

b)2/5+3/5:(-3/2)=1/2

=2/3+2/5+1/2

=16/15+1/2

=47/30

c)1 và 2/5 - (-1/2)^2 + 7/10

=7/5 - 1/4 + 7/10

=23/20 + 7/10

=37/20

\(a,2^2+4^2+6^2+...+20^2\)

\(=1^2.2^2+2^2.2^2+3^2.2^2+...+10^2.2^2\)

\(=\left(1^2+2^2+3^2+...+10^2\right).2^2\)

\(=385.4\)

\(=1540\)

\(b,\left(0.25\right)^2+\left(0.5\right)^2+...+\left(2.5\right)^2\)

\(=1^2.0,25^2+2^2.0,25^2+...+10^2.0,25^2\)

\(=\left(1^2+2^2+...+10^2\right).0,25^2\)

\(=385.0.0625\)

\(=24.0625\)

a/ Có tự ghi lại đề

= ((2+8)+(4+6)+10+(12+18)+(14+16)+20))^2

= ((10+10+10+30+30+20))^2

=(110)^2

=100^2+10^2

=10000+100

=10100