5,Ta có

A=1/2+1/2^2+1/2^3+...+1/2^100

2A=1+1/2+1/2^2+1^2/3+...+1/2^99

2A-A=(1+1/2+1/2^2+1^2/3+...+1/2^99)-(1/2+1/2^2+1/2^3+...+1/2^100)

A=1-1/2^100

Ta có : A = 1.2.3 + 2.3.4 + 4.5.6 + ..... + 98.99.100

=> 6A = 1.2.3.4 - 1.2.3.4 + 2.3.4.5 - 2.3.4.5 + ...... + 98.99.100.101

=> 6A = 98.99.100.101 

=> A = \(\frac{98.99.100.101}{6}=16331700\)

có 2017² đồng dư 1 mod (3)
   => (2017²)⁵⁰ đồng dư 1 mod (3)
=> (2017²)⁵⁰-1 đồng dư 1-1 = 0 mod (3)
=> dpcm

Câu a)
\(A=2^{100}-2^{99}+2^{98}-2^{97}+...+2^2-2\)
\(=\left(2^{100}+2^{99}+2^{98}+2^{97}+...+2^2+2\right)-2\left(2^{99}+2^{97}+2^{95}+...+2^3+2\right)\)
\(=\left(2^{100}+2^{99}+2^{98}+2^{97}+...+2^2+2\right)-\left(2^{100}+2^{98}+2^{96}+...+2^4+2^2\right)\)
\(=2^{99}+2^{97}+2^{95}+...+2^3+2\)
\(=\frac{2^2\cdot\left(2^{99}+2^{97}+2^{95}+...+2^3+2\right)-\left(2^{99}+2^{97}+2^{95}+...+2^3+2\right)}{3}\)
\(=\frac{\left(2^{101}+2^{99}+2^{97}+...+2^5+2^3\right)-\left(2^{99}+2^{97}+2^{95}+...+2^3+2\right)}{3}\)
\(=\frac{2^{101}-2}{3}\)

\(2B=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{2015.2016.2017}\)

\(2B=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{2.4}+...+\frac{1}{2015.2016}-\frac{1}{2016.2017}\)

\(2B=\frac{1}{1.2}-\frac{1}{2016.2017}\)

\(B=\frac{\frac{1}{1.2}-\frac{1}{2016.1017}}{2}\)

Tính 

a,1.2.3+2.3.4+3.4.5+......+ 98.99.100

b,1 bình +2 bình +3 bình +....+100 bình

Giải:Đặt A=1.2.3+2.3.4+..........+98.99.100

4A=1.2.3.4+2.3.4.5-1.2.3.4+...........+98.99.100.101-97.98.99.100

4A=98.99.100.101=97990200\(\Rightarrow A=24497550\)

b,Đặt B=1²+2²+................+100²

B=1.(2-1)+2.(3-1)+.............+100.(101-1)

B=1.2+2.3+.......+100.101-1-2-..........-100

Đặt C=1.2+2.3+........+100.101

3C=1.2.3+2.3.4-1.2.3+........+100.101.102-99.100.101

3C=100.101.102=1030200\(\Rightarrow C=343400\)

\(\Rightarrow B=343400-\frac{100.101}{2}=343400-5050=338350\)

Ngu như con bò

vay sao chi

a)S có số số hạng là:

(100-1):1+1=100(số hạng)

Vậy S bằng:

(1+100)X100:2=5050

b)S có số số hạng là:

(998-2):2+1=499(số hạng)

Vậy S bằng:

(2+998)X499:2=249500

c)S=1.2+2.3+3.4+...+98.99+99.100

3S=1.2.3+2.3.3+3.4.3+...+98.99.3+99.100.3

3S=1.2.3+2.3.(4-1)+3.4.(5-2)+...+98.99.(100-97)+99.100.(101-98)

3S=1.2.3+(2.3.4-1.2.3)+(3.4.5-2.3.4)+...+(98.99.100-97.98.99)+(99.100.101-98.99.100)

3S=(1.2.3+2.3.4+3.4.5+...+98.99.100+99.100.101)-(1.2.3+2.3.4+3.4.5+...+97.98.99+98.99.100)

3S=99.100.101=999900

S=333300

d)S=1.2.3+2.3.4+...+98.99.100

4S=1.2.3.4+2.3.4.4+...+98.99.100.4

4S=1.2.3.4+2.3.4.(5-1)+...+98.99.100.(101-97)

4S=1.2.3.4+(2.3.4.5-1.2.3.4)+...+(98.99.100.101-97.98.99.100)

4S=(1.2.3.4+2.3.4.5+...98.99.100,101)-(1.2.3.4+2.3.4.5+...+97.98.99.100)

4S=98.99.100.101=97990200

S=24497550

e)\(S=1^2+2^2+3^2+...+99^2+100^2\)

\(S=1.\left(2-1\right)+2.\left(3-1\right)+3.\left(4-1\right)+...+99.\left(100-1\right)+100.\left(101-1\right)\)

\(S=\left(1.2-1\right)+\left(2.3-2\right)+\left(3.4-3\right)+...+\left(99.100-99\right)+\left(100.101-100\right)\)

\(S=\left(1.2+2.3+...+99.100+100.101\right)-\left(1+2+3+...+99+100\right)\)

Dựa vào kết quả câu a và c ta được:

S=333300-5050=328250