chi tiết hơn được ko bạn?

\(A=1\cdot2+2\cdot3+...+151\cdot152\)

\(=1\left(1+1\right)+2\left(1+2\right)+...+151\left(1+151\right)\)

\(=\left(1+2+3+...+151\right)+\left(1^2+2^2+...+151^2\right)\)

\(=\dfrac{151\left(151+1\right)}{2}+\dfrac{151\left(151+1\right)\left(2\cdot151+1\right)}{6}\)

\(=151\cdot76+\dfrac{151\cdot152\cdot303}{6}\)

\(=151\cdot76+151\cdot7676=1170552\)

\(C=2\cdot4+4\cdot6+...+2024\cdot2026\)

\(=2\cdot2\left(1\cdot2+2\cdot3+...+1012\cdot1013\right)\)

\(=4\left[1\left(1+1\right)+2\left(1+2\right)+...+1012\left(1+1012\right)\right]\)

\(=4\left[\left(1+2+...+1012\right)+\left(1^2+2^2+...+1012^2\right)\right]\)

\(=4\left[1012\cdot\dfrac{1013}{2}+\dfrac{1012\left(1012+1\right)\left(2\cdot1012+1\right)}{6}\right]\)

\(=4\left[506\cdot1013+345990150\right]\)

\(=1386010912\)

\(M=1^2+2^2+...+2024^2\)

\(=\dfrac{2024\left(2024+1\right)\cdot\left(2\cdot2024+1\right)}{6}\)

\(=2024\cdot2025\cdot\dfrac{4049}{6}\)

=2765871900

\(N=1^3+2^3+...+100^3\)

\(=\left(1+2+3+...+100\right)^2\)

\(=\left[\dfrac{100\left(100+1\right)}{2}\right]^2\)

\(=\left[50\cdot101\right]^2=5050^2\)

\(Q=1^3+2^3+...+2024^3\)

\(=\left(1+2+3+...+2024\right)^2\)

\(=\left[\dfrac{2024\left(2024+1\right)}{2}\right]^2\)

\(=\left[1012\left(2024+1\right)\right]^2\)

\(=2049300^2\)

BÀI 1:

\(S=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}\)

\(S=1+\frac{1}{1.2}+\frac{1}{2.2}+\frac{1}{2.4}+\frac{1}{4.4}+\frac{1}{4.8}\)

\(S=1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{4}+\frac{1}{4}-\frac{1}{8}\)

\(S=1+1-\frac{1}{8}\)

\(S=\frac{15}{8}\)

BÀI 2:

\(A=1.2+2.3+3.4+...+98.99\)

\(\Rightarrow3A=1.2.3+2.3.3+3.4.3+...+98.99.3\)

\(3A=1.2.\left(3-0\right)+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+98.99.\left(100-97\right)\)

\(3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+98.99.100-97.98.99\)

\(3A=\left(1.2.3+2.3.4+3.4.5+98.99.100\right)-\left(1.2.3+2.3.4+...+97.98.99\right)\)

\(3A=98.99.100\)

\(3A=970200\)

\(\Rightarrow A=970200:3\)

\(A=323400\)

CHÚC BN HỌC TỐT!!!
 

A = 1.2 + 2.3 + 3.4 + ...+ 59.60

3A = 1.2.3 + 2.3.3 + 3.4.3 + ...+ 59.60.3

3A = 1.2.(3-0) + 2.3.(4-1) + 3.4.(5-2) +...+ 59.60.(61-58)

3A = 1.2.3 - 0 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 +...+ 59.60.61 - 58.59.60

3A = 58.59.60 => A = 58.59.60 : 3 = 68 440

B = 1² + 2² + 3² + 59²

B = 1.2 + 2.2 + 3.3 + 59.59

B = 2 + 4 + 9 + 3481

B = 3496

vậy A - B = 68 440 - 3 496 = 64 944

( bấm nhé )

B= bạn k ghi cái dấu ba chấm à

Ta có : S = 1.2 + 2.3 + 3.4 + ..... + 32.33

=> 3S = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + ...... + 32.33.34

=> 3S = 32.33.34

=> S = \(\frac{32.33.34}{3}=11968\)

a) Đặt A = 1/2 + 1/4 + 1/8 + 1/16 + 1/32

A = 1/2 + 1/2² + 1/2³ + 1/2⁴ + 1/2⁵

2A = 2(1/2 + 2² + 1/2³ + 1/2⁴ + 1/2⁵)

2A = 1 + 1/2 + 1/2² + 1/2³ + 1/2⁴

2A - A = (1 + 1/2 + 1/2²  + 1/2³ + 1/2⁴) - (1/2  + 1/2²  + 1/2³ + 1/2⁴ + 1/2⁵)

A = 1 - 1/2⁵

A = 31/32

b) 2/1.2 + 2/2.3 + 2/3.4 + ... + 2/18 . 19 + 2/19.20

= 2(1/1.2  + 1/2.3  + 1/3.4 + ... + 1/18.19 + 1/19.20)

= 2.(1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/18 - 1/19 + 1/19 - 1/20)

= 2. (1 - 1/20)

= 2.19/20

= 19/10

bạn thiếu ý C 

cau hỏi tương tự ko có mà!!!!!!!!!!!!!!!!!!!!!!!!!!!!

3C=1.2.3+2.3.(4-1)+3.4.(5-2)+...+2014.2015.(2016-2013)

3C=2014.2015.2016

C=2014.2015.2016:3

ta có :

\(a-b=1.2+\left(2.3-2^2\right)+\left(3.4-3^2\right)+..+\left(98.99-98^2\right)\)

\(=2+2+3+4+..+98\)

\(=1+\left(1+2+3+..+98\right)=1+98\times\frac{99}{2}=4852\)

 A = 1.2 + 2.3 + 3.4 + ... + 2013.2014 
3A = 1.2.3 + 2.3.3 + 3.4.3 + ... + 2013.2014.3 
Mà : 
1.2.3 = 1.2.3 
2.3.3 = 2.3.4 - 2.3.1 
3.4.3 = 3.4.5 - 3.4.2 

2012.2013.3 = 2012.2013.2014 - 2012.2013.2011 
2013.2014.3 = 2013.2014.2015 - 2013.2014.2012 
Cộng tất cả, vế theo vế ---> 3S = 2013.2014.2015 
---> A = 2013.2014.2015 / 3 = 2723058910.

của bạn đây

C=1*2+2*3+3*4+...+98*99

C=2+6+12+...+9702

C=2+9702

C=9704

vay C=9704

D=(1*99+2*99+3*99+...+99*99)-(1*2+2*3+3*4+...+98*99)

D=(99+198+297+...+9801)-(2+6+12+...+9702)

D=(99+9801)-(2+9702)

D=9900-9704

D=196

vay D=196

ai di qua dong tinh thi nho h cho minh nhe