Đặt \(S=\frac{1}{4}+\frac{1}{12}+\frac{1}{36}+...+\frac{1}{2010}+\frac{1}{6030}.\)

        \(\Rightarrow3S=\frac{3}{4}+\frac{3}{12}+\frac{3}{36}+...+\frac{3}{2010}+\frac{3}{6030}\)

                   \(=\frac{3}{4}+\frac{1}{4}+\frac{1}{12}+...+\frac{1}{670}+\frac{1}{2010}\)

  \(\Rightarrow3S-S=2S=\left(\frac{3}{4}+\frac{1}{4}+\frac{1}{12}+...+\frac{1}{670}+\frac{1}{2010}\right)-\left(\frac{1}{4}+\frac{1}{12}+\frac{1}{36}+...+\frac{1}{2010}+\frac{1}{6030}\right)\)

\(2S=\frac{3}{4}-\frac{1}{6030}\)

\(\Rightarrow S=\frac{\frac{3}{4}-\frac{1}{6030}}{2}\)

đặt A=1/4+1/12+1/36+........+1/6030

3A=1+1/4+1/12+.........+1/2010

-2A=1/6030-1

A=(1/6030-1)/-2

Đặt A= 1/4+1/12+1/36+1/108+1/324 +1/972

            =1/4.(1+1/3+1/9+1/27+1/81+1/243)

            =1/4(1+1/3 +1/3^2 +1/3^3 +1/3^4 +1/3^5)

             gọi M=1+1/3+1/3^2+1/3^3+1/3^4+1/3^5

                     3M=3+1+1/3+1/3^2+1/3^3+1/3^4

                     3M-M=(3+1+1/3+1/3^2+1/3^3+1/3^4)-(1+1/3+1/3^2+1/3^3+1/3^4+1/3^5)

                         2M=3-1/3^5

                         M=3/2-1/3^5.2  

                A=1/4.(3/2-1/3^5.2)

                A=3/8-1/1944=91/243

                  Vậy .......

                Chúc bạn học tốt .....^-^      

Bài làm

1.Tính nhanh

(2+4+6+8+...+150).(36.333-108.111)

=(2+4+6+...+150).(11 988-11 988)

=(2+4+6+...+150).0

=0

=19991999.1998-19991999.1998

=0

a, (2 + 3 + 4 + ... + 150)(36.333 - 108.111)

= (2 + 3 + 4 + ... + 150)(36.3.111 - 36.3.111)

= (2 + 3 + 4 + ... + 150).0

= 0

b, 19991999.1998 - 19981999.1998

= 1998(19991999 - 19981999)

= 1998.10000

= 19980000

c, A = 1.2 + 2.3 + 3.4 + ... + 9.10

3A = 1.2.3 + 2.3.3 + 3.4.3 + ... + 9.10.3

3A = 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 9.10.(11 - 8)

3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 9.10.11 - 8.9.10

3A = 9.10.11

3A = 990

A = 990 : 3 = 330 

1)  \(\left(2+4+6+...+150\right).\left(36.33-108.111\right)\)

\(=\left(2+4+6+...+150\right).\left(108.111-108.111\right)\)

\(=\left(2+4+6+...+150\right).0\)

\(=0\)

\(19991999.1998-19981998.1999\)

\(=1999.10001.1998-1998.10001.1999\)

\(=0\)

\(A=1.2+2.3+...+9.10\)

\(\Rightarrow3A=1.2.3+2.3.3+...+9.10.3\)

\(=1.2.3+2.3.\left(4-1\right)+...+9.10.\left(11-8\right)\)

\(=1.2.3+2.3.4-1.2.3+...+9.10.11-8.9.10\)

\(=9.10.11\)

\(\Rightarrow A=\frac{9.10.11}{3}=330\)

(1+1/2).(1+1/3).(1+1/4).....(1+1/2010).(1+1/2011)

=3/2.4/3.5/4...2011/2010.2012/2011

=2012/2=1006

 Đặt tổng trên là A ta có 
A/3=1/12+1/36+1/108+ +1/2916+1/8748 
A-A/3=2A/3= 
=1/4-1/8748=2186/8748 
=>A=1093/1916

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