`#3107.101107`

1.

a)

`1/(1*4) + 1/(4*7) + 1/(7*10) + ... + 1/(100*103)`

`= 1/3 * (3/(1*4) + 3/(4*7) + 3/(7*10) + ... + 3/(100*103) )`

`= 1/3 * (1 - 1/4 + 1/4 - 1/7 + ... + 1/100 - 1/103)`

`= 1/3* (1 - 1/103)`

`= 1/3*102/103`

`= 34/103`

b)

`-1/3 + (-1/15) + (-1/35) + (-1/63) + ... + (-1/9999)`

`= - 1/3 - 1/15 - 1/35 - 1/63 - ... - 1/9999`

`= - (1/3 + 1/15 + 1/35 + ... + 1/9999)`

`= - (1/(1*3) + 1/(3*5) + 1/(5*7) + ... + 1/99*101)`

`= - 1/2 * (2/(1*3) + 2/(3*5) + 2/(5*7) + ... + 2/99*101)`

`= - 1/2* (1 - 1/3 + 1/3 - 1/5 + ... + 1/99 - 1/101)`

`= -1/2 * (1 - 1/101)`

`= -1/2*100/101`

`= -50/101`

2.

`3/(1*4) + 3/(4*7) + ... + 3/(94*97) + 3/(97*100)`

`= 1 - 1/4 + 1/4 - 1/7 + ... + 1/94 - 1/97 + 1/97 - 1/100`

`= 1-1/100`

`= 99/100`

S = 1.4 + 4.7 + 7.10 + 10.13 + ... + 61.64

1.4.9 = 1.4.(7 + 2) = 1.4.7 + 1.4.2

4.7.9 = 4.7.(10 - 1) = 4.7.10 - 1.4.7

7.10.9 = 7.10.(13 - 4) = 7.10.13 - 4.7.10

10.13.9 = 10.13.(16 - 7) = 10.13.16 - 7.10.13

.......................................................................

61.64.9 = 61.64.(67 - 58) = 61.64.67 - 58.61.64

Cộng vế với vế ta có:

1.4.9 + 4.7.9 + 7.10.9 +...+ 61.64.9 = 1.4.2 + 61.64.67

9(1.4 + 4.7 + 7.10+ ...+ 61.64) = 261576

  1.4 + 4.7 + 7.10 +...+ 61.64 = 261576 : 9

1.4 + 4.7 + 7.10 + ... + 61.64 = 29064 

\(A=\dfrac{3}{1\cdot4}+\dfrac{3}{4\cdot7}+\dfrac{3}{7\cdot10}+...+\dfrac{3}{61\cdot64}+\dfrac{3}{64\cdot67}\)

\(A=1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{61}-\dfrac{1}{64}+\dfrac{1}{64}-\dfrac{1}{67}\)

\(A=1-\dfrac{1}{67}\) < 1

=> A<1

Ta có:

\(A=\dfrac{3}{1.4}+\dfrac{3}{4.7}+\dfrac{3}{7.10}+...+\dfrac{3}{61.64}+\dfrac{3}{64.67}\)

\(=3.\dfrac{1}{3}.\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{61}-\dfrac{1}{64}+\dfrac{1}{64}-\dfrac{1}{67}\right)\)

\(=3.\left(1-\dfrac{1}{67}\right)\)

\(=3.\dfrac{66}{67}\)

\(=\dfrac{198}{67}\)

Vì \(\dfrac{198}{67}\) có tử lớn hơn mẫu nên \(\dfrac{198}{67}>1\)

Vậy \(A>1\)

\(C=\frac{2}{3}.\left(\frac{3}{1.4}+\frac{3}{4.7}+........+\frac{3}{61.64}\right)\)

\(=\frac{2}{3}.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+.........+\frac{1}{61}-\frac{1}{64}\right)\)

\(=\frac{2}{3}.\left(1-\frac{1}{64}\right)\)

\(=\frac{2}{3}.\frac{63}{64}\)

\(=\frac{21}{32}\)

\(C=\frac{2}{1.4}+\frac{2}{4.7}+\frac{2}{7.10}+...+\frac{2}{61.61}\)

\(=2.\frac{1}{3}.\left(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{61.64}\right)\)

\(=\frac{2}{3}.\left(\frac{1}{1}-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{61}-\frac{1}{64}\right)\)

\(=\frac{2}{3}.\left(1-\frac{1}{64}\right)\)

\(=\frac{2}{3}.\frac{63}{64}\)

\(=\frac{21}{32}\)

\(=-\left(\dfrac{1}{1}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+...+\dfrac{1}{61}-\dfrac{1}{64}\right)=-\dfrac{1}{63}\)

a) 1x( 1+3) ; 4x( 4+3) ; 7 x ( 7+3) ; 10 x ( 10 +3) ; ,,,,,,,,

b)1720

st1 = 1.4 = [ 3.(1-1) + 1].[3.(1-1) + 4]

st2 = 4.7 = [3.(2-1) + 1].[3.(2-1) + 4]

st3 = 7.10 = [3.(3-1) + 1].[3.(3-1) + 4]

..........................................................

stn =          [3.(n - 1) + 1].(3.(n - 1) + 4]

stn = (3n - 2).(3n + 1)

số hạng thứ 15 của dãy số trên là: 

     (3.15 - 2).(3.15 + 1) = 43.46 = 1978

1/1.4+1/4.7+1/7.10+...+1/16.19

=[1/1.4+1/4.7+1/7.10+...+1/16.19] x 3

= 3/1.4+3/4.7+3/7.10+...+3/16.19

= 1-1/4+1/4-1/7+1/7-1/10+....+1/16-1/19

=1-1/19

=18/19 :3

=6/19

ĐÂY BẠN NHÉ CHÚC BẠN HỌC TỐT NHỚ K CHO MÌNH

A = \(\dfrac{1}{10}+\dfrac{1}{15}+...+\dfrac{1}{120}\)

A= \(\dfrac{2}{2}.\left(\dfrac{1}{10}+\dfrac{1}{15}+...+\dfrac{1}{120}\right)\) 

A= \(2.\left(\dfrac{1}{20}+\dfrac{1}{30}+...+\dfrac{1}{240}\right)\)

A= \(2.\left(\dfrac{1}{4.5}+\dfrac{1}{5.6}+...+\dfrac{1}{15.16}\right)\) 

A=\(2.\left(\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+...+\dfrac{1}{15}-\dfrac{1}{16}\right)\)

A=\(2.\left(\dfrac{1}{4}-\dfrac{1}{16}\right)\) 

A=\(\dfrac{2.3}{16}\) 

A= \(\dfrac{3}{8}\)

f,F=3. (1/2 .3 + 1/3.4 +...+ 1/99.100)

    = 3. (1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 +...+ 1/99 - 1/100

    = 3. (1/2 - 1/100)

    = 3. 49/100

    = 147/100

g, G = 5/3. (3/1.4 + 3/4.7 +...+ 3/61.64)

        = 5/3 . (1 - 1/4 + 1/4 - 1/7 +...+ 1/61 - 164

        = 5/3 . (1-1/64)

        = 5/3 . 63/64

        = 105/64