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 

3/1.4+3/4.7+3/7.10+3/10.13

=1 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + 1/10 - 1/13

=1 - 1/13

=12/13

#)Trả lời :

Gọi tổng trên là A 

\(A=\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+\frac{3}{10.13}\)

\(A=3-\frac{3}{4}+\frac{3}{4}-\frac{3}{7}+\frac{3}{7}-\frac{3}{10}+\frac{3}{10}-\frac{3}{13}\)

\(A=3-\frac{3}{13}\)

\(A=2\frac{10}{13}=\frac{36}{13}\)

               #~Will~be~Pens~#

\(\frac{3}{1\cdot4}+\frac{3}{4\cdot7}+\frac{3}{7\cdot10}+\frac{3}{10\cdot13}\)

\(=3-\frac{3}{4}+\frac{3}{4}-\frac{3}{7}+\frac{3}{7}-\frac{3}{10}+\frac{3}{10}-\frac{3}{13}\)

\(=3-\frac{3}{13}\)

\(=\frac{36}{13}\)

\(\dfrac{3}{1\cdot4}+\dfrac{3}{4\cdot7}+\dfrac{3}{7\cdot10}+\dfrac{3}{10\cdot13}+\dfrac{3}{13\cdot16}\)

\(=\dfrac{3\cdot1}{1\cdot4}+\dfrac{3\cdot1}{4\cdot7}+\dfrac{3\cdot1}{7\cdot10}+\dfrac{3\cdot1}{10\cdot13}+\dfrac{3\cdot3}{13\cdot16}\)

\(=3\cdot\left(\dfrac{1}{1\cdot4}+\dfrac{1}{4\cdot7}+\dfrac{1}{7\cdot10}+\dfrac{1}{10\cdot13}+\dfrac{1}{13\cdot16}\right)\)

\(=3\cdot\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{13}+\dfrac{1}{13}-\dfrac{1}{16}\right)\)

\(=3\cdot\left(1-\dfrac{1}{16}\right)\)

\(=3\cdot\left(\dfrac{16}{16}-\dfrac{1}{16}\right)\)

\(=3\cdot\dfrac{15}{16}\)

\(=\dfrac{45}{16}\)

ngu :3

vcl

  3/1.4 + 3/4.7 + .. +3/13.16

= 1/1 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + 1/10 - 1/13 + 1/13 - 1/16

= 1/1 - 1/16

= 15/16

\(=\frac{15}{16}\)

đúng cho mk nha Minh Thư Nguyễn

x/1.4+x/4.7+x/7.10+x/10.13+x/13.16=5/2

=>x/3(1/4-1/7+1/7-1/10+1/10-1/13+1/13-1/16)=5/2

=>x/3.(1/4-1/16)=5/2

=>x/3.3/16=5/2

=>x/3=5/2:3/16

=>x/3=40/3

=>x=40

Vậy x=40

x/1.4 + x/4.7 + x/7.10 + x/10.13 + x/13.16 = 5/6

=> x.1/3.(3/1.4 + 3/4.7 + 3/7.10 + 3/10.13 + 3/13.16) = 5/6

=> x.1/3.(1 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + 1/10 - 1/13 + 1/13 - 1/16) = 5/6

=> x.1/3.(1 - 1/16) = 5/6

=> x.1/3.15/16 = 5/6

=> x.1/3 = 5/6 : 15/16

=> x.1/3 = 8/9

=> x = 8/9 : 1/3

=> x = 8/3

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

ta nhân 3 cả hai vế, được : 

\(\left(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{102.105}\right)x=3\)

hay 

\(\left(\frac{4-1}{1.3}+\frac{7-4}{4.7}+...+\frac{105-102}{102.105}\right)x=3\) \(\Leftrightarrow\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+..+\frac{1}{102}-\frac{1}{105}\right)x=3\)

\(\Leftrightarrow\left(1-\frac{1}{105}\right)x=3\Leftrightarrow\frac{104}{105}.x=3\Leftrightarrow x=\frac{315}{104}\)

4/1.4+4/4.7+4/7.10+4/10.13

= 4/3(3/1.4+3/4.7+3/7.10+3/10.13)

=4/3(1/1-1/4+1/4-1/7+1/7-1/10+1/10-1/13)

=4/3(1/1-1/13)

=4/3.12/13

=16/13