Ta có : S₁ = 1 + (-3) + 5 + (-7) + .... + 17

                = (1 - 3) + (5 - 7) + (9 - 11)+ (13 - 15) + 17 

                = -2 + -2 + -2 + -2 + 17 

                = -2 x 4 + 17 

                = -8 + 17 

             S₁ = 9

S₂ = (4 - 2) + (8 - 6) + (12 - 10) + (16 - 14) + -18

     = 2 x 4 - 18 

S₂ = -10

S₁ + S₂ = 9 - 10 = -1

S₁=1+(-3)+5+(-7)+...+17.

S₁=-2+(-2)+....+(-2).(9 số -2).

S₂=-2+4+(-6)+....+(-18)

S₂=-2+(-2)+...+(-2).(9 số -2).

=> (-2).(9+9)=-36.

S₁ = 1 + (-3) + 5 + (-7) + ............+ 17

= [ 1+(-3) ] + [5+(-7) + .........+ 17

= -2   +  ( -2 ) + ......+ 17  ( có 4 số -2)

= -2  . 4  + 17

=  -8  + 17 = 9

S₂ = -2  + 4 + (-6) +8 ....+(-18)

= [ (-2)+4]  + [ (-6) + 8 + ......+ (-18) (có 4 cặp số )

= 2  + 2 + .....+ (-18)

= 2 . 4 + (-18)

= 8 + (-18)

= -10

suy ra S₁ + S₂  = 9 + (-10) = -1

S = 1 * 2 + 2 * 3 + 3 * 4 + ...... + 28 * 29 + 29 * 30

3S = 1 x 2 x 3 - 1 x 2 x 3 + 2 x 3 x 4 - 2 x 3 x 4 + .............. + 29 x 30 x 31

3S = 29 x 30 x 31

S = 29 x 30 x 31 : 3 = 8990

Ta có : S = 1.2 + 2.3 + 3.4 + ..... + 28.29 + 29.30

<=> 3S = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + ...... + 29.30.31

<=> 3S = 29.30.31 

<=> S = 29.30.31 / 3 = 8990

Chúc học tốt nhé !

s = 1.2 + 2.3 + 3.4 +...+ 29.29 + 29.30

=> 3S = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 +...+ 29.30.31

3S = 29.30.31

S = 29.30 . 31/3 = 8990

\(S=1+3+3^2+3^3+3^4+.....+3^{16}\)

\(=\left(1+3+3^2\right)+\left(3^3+3^4+3^5\right)+.....+\left(3^{2014}+3^{2015}+3^{2016}\right)\)

\(=1\left(1+3+3^2\right)+3^3\left(1+3+3^2\right)+......+3^{2014}\left(1+3+3^2\right)\)

\(=1.13+3^3.13+.....+3^{2014}.13\)

\(=13\left(1+3^3+....+3^{2014}\right)⋮13\)

\(\Rightarrow S⋮13\)

\(S=7+7^2+7^3+...7^{20}\)

Ta có: \(7S=7.\left(7+7^2+7^3+...+7^{20}\right)\)

\(7S=7^2+7^3+7^4+...+7^{21}\)

\(7S-S=\left(7^2+7^3+7^4+...+7^{21}\right)-\left(7+7^2+7^3+...+7^{20}\right)\)

\(6S=\left(7^{21}-7\right)\)

\(S=\left(7^{21}-7\right):6\)

Chúc bạn học tốt

7S=7^2+7^3+...+7^21

=>6S=7^21-7

=>S=(7^21-7)/6

A = 1x2 + 2x3 + 3x4 + ……….. + 29 x30

A x 3 = 1x2x3 + 2x3x3 + 3x4x3 + ……… + 29x30x3 Ax3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) +………… 29x30x(31-28)

Ax3 = 1x2x3 + 2x3x4 -1x2x3 + 3x4x5 – 2x3x4 +……..+ 29x30x31 – 28x29x30 Ax3 = 29x30x31

A = 29x30x31 : 3

A = 290x31

A = 1x2 + 2x3 + 3x4 + ……….. + 29 x30
A x 3 = 1x2x3 + 2x3x3 + 3x4x3 + ……… + 29x30x3
Ax3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) +………… 29x30x(31-28)
Ax3 = 1x2x3 + 2x3x4 -1x2x3 + 3x4x5 – 2x3x4 +……..+ 29x30x31 – 28x29x30
Ax3 = 29x30x31
A = 29x30x31 : 3
A = 290x31

\(S=\dfrac{1}{1\cdot3}+\dfrac{1}{3\cdot5}+\dfrac{1}{5\cdot7}+\dfrac{1}{7\cdot9}-\left(\dfrac{1}{2\cdot4}+\dfrac{1}{4\cdot6}+\dfrac{1}{6\cdot8}+\dfrac{1}{8\cdot10}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+\dfrac{2}{7\cdot9}\right)-\dfrac{1}{2}\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+\dfrac{2}{6\cdot8}+\dfrac{2}{8\cdot10}\right)\)

\(=\dfrac{1}{2}\left(1-\dfrac{1}{9}\right)-\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{10}\right)\)

\(=\dfrac{1}{2}\cdot\dfrac{8}{9}-\dfrac{1}{2}\cdot\dfrac{2}{5}\)

\(=\dfrac{4}{9}-\dfrac{1}{5}\)

\(=\dfrac{11}{45}\)

Ta có:

\(T\left(-2\right)=a_0-2a_1+2^2a_2-...-2^{29}a_{29}+2^{30}a_{30}=a_0+H=\left(1+4\right)^{15}\)

\(\Leftrightarrow1+H=5^{15}\)

\(\Leftrightarrow H=5^{15}-1\)