a) A = \(\dfrac{1}{2}\)+\(\dfrac{5}{6}\)+\(\dfrac{11}{12}\)+\(\dfrac{19}{20}\)+\(\dfrac{29}{30}\)+\(\dfrac{41}{42}\)+\(\dfrac{55}{56}\)+\(\dfrac{71}{72}\)+\(\dfrac{89}{90}\)
b) B = 15 + 2\(^4\) + 2\(^5\) + 2\(^6\) + ... + 2\(^{2020}\)
c) (\(\dfrac{1}{1.2.3}\) + \(\dfrac{1}{2.3.4}\) + ... + \(\dfrac{1}{8.9.10}\)) . x = \(\dfrac{44}{45}\)
d) \(\dfrac{x}{6}\) - \(\dfrac{2}{y}\) = \(\dfrac{1}{30}\)
e) E = 1 + \(\dfrac{1}{2}\) + \(\dfrac{1}{3}\) + \(\dfrac{1}{4}\) + ... + \(\dfrac{1}{2^{100}-1}\); Chứng minh 50 < x < 100
`a,A=1/2+5/6+11/12+19/20+29/30+41/42+55/56+71/72+89/90`
`A=1-1/2+1-1/6+1-1/12+1-1/20+1-1/30+1-1/42+1-1/56+1-1/72+1-1/90`
`A=(1+1+1+1+1+1+1+1+1)-(1/2+1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90)`
`A=9-(1/(1*2)+1/(2*3)+1/(3*4)+1/(4*5)+1/(5*6)+1/(6*7)+1/(7*8)+1/(8*9)+1/(9*10))`
`A=9-(1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10)`
`A=9-1+1/10`
`A=8+1/10`
`A=81/10`
`b,B=15+2^4+...+2^2020`
`B=1+2+2^2+2^3+2^4+...+2^2020`
`2B=2+2^2+2^3+2^4+...+2^2021`
`2B-B=(2+2^2+2^3+2^4+...+2^2021)-(1+2+2^2+2^3+...+2^2020)`
`B=2^2021-1`
`c) (1/(1.2.3) + 1/(2.3.4) + ... + 1/(8.9.10)) . x = 44/45`
`=> (2/(1.2.3) + 2/(2.3.4) + ... + 2/(8.9.10)) . x = 88/45`
`=( 1/(1.2) - 1/(2.3) + 1/(2.3) - 1/(3.4) + ... + 1/(8.9) - 1/(9.10)).x = 88/45`
`= (1/2 - 1/90) . x = 88/45`
`=> 22/45 . x = 88/45`
`=> x = 4`
