a.

$S=1+2+2^2+2^3+...+2^{2017}$
$2S=2+2^2+2^3+2^4+...+2^{2018}$

$\Rightarrow 2S-S=(2+2^2+2^3+2^4+...+2^{2018}) - (1+2+2^2+2^3+...+2^{2017})$

$\Rightarrow S=2^{2018}-1$

b.

$S=3+3^2+3^3+...+3^{2017}$
$3S=3^2+3^3+3^4+...+3^{2018}$

$\Rightarrow 3S-S=(3^2+3^3+3^4+...+3^{2018})-(3+3^2+3^3+...+3^{2017})$

$\Rightarrow 2S=3^{2018}-3$
$\Rightarrow S=\frac{3^{2018}-3}{2}$
 

Câu c, d bạn làm tương tự a,b. 

c. Nhân S với 4. Kết quả: $S=\frac{4^{2018}-4}{3}$

d. Nhân S với 5. Kết quả: $S=\frac{5^{2018}-5}{4}$

a) \(S=1+2+2^2+..+2^{2022}\)

\(2S=2+2^2+2^3+...+2^{2023}\)

\(2S-S=2+2^2+2^3+...+2^{2023}-1-2-2^2-...-2^{2022}\)

\(S=2^{2023}-1\)

b) \(S=3+3^2+3^3+...+3^{2022}\)

\(3S=3^2+3^3+...+3^{2023}\)

\(3S-S=3^2+3^3+....+3^{2023}-3-3^2-...-3^{2022}\)

\(2S=3^{2023}-3\)

\(\Rightarrow S=\dfrac{3^{2023}-3}{2}\)

c) \(S=4+4^2+4^3+...+4^{2022}\)

\(4S=4^2+4^3+...+4^{2023}\)

\(4S-S=4^2+4^3+...+4^{2023}-4-4^2-...-4^{2022}\)

\(3S=4^{2023}-4\)

\(S=\dfrac{4^{2023}-4}{3}\)

d) \(S=5+5^2+...+5^{2022}\)

\(5S=5^2+5^3+...+5^{2023}\)

\(5S-S=5^2+5^3+...+5^{2023}-5-5^2-...-5^{2022}\)

\(4S=5^{2023}-5\)

\(S=\dfrac{5^{2023}-5}{4}\)

\(A=\frac{7}{3\times13}+\frac{7}{13\times23}+...+\frac{7}{53\times63}\)

\(A=\frac{7}{10}.\left[\left(\frac{1}{3}-\frac{1}{13}\right)+\left(\frac{1}{13}-\frac{1}{23}\right)+....+\left(\frac{1}{53}-\frac{1}{63}\right)\right]\)

\(A=\frac{7}{10}.\left(\frac{1}{3}-\frac{1}{13}+\frac{1}{13}-\frac{1}{23}+....+\frac{1}{53}-\frac{1}{63}\right)\)

\(A=\frac{7}{10}.\left(\frac{1}{3}-\frac{1}{63}\right)\)

\(A=\frac{7}{10}.\frac{20}{63}\)

\(A=\frac{2}{9}\)

huỳnh thị ngân hà, nau te, trần như tính sao ra z? 

S = 13+10+23+20+33+30+...+103+100

S =  13+23+33+...+103+10.100

S = 3025+1000

S = 4025

6020 , chắc chắn đúng 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000%

A=7*(1/3*13+1/13*23+1/23*33+1/33*43+1/43*53+1/53*63)

A=7/10(1/3-1/13+1/13-1/23+1/23-1/33+1/33-1/43+1/43-1/53+1/53-1/63)

A=7/10*(1/3-1/63)

A=7/10*20/63

A=2/9

\(B=-\left(1+4+7+...+100\right)\\ B=-\dfrac{\left(100+1\right)\left[\left(100-1\right):3+1\right]}{2}=-\dfrac{101\cdot34}{2}=-1717\\ C=10+10+10+10-103=50-103=-53\)

e,1³ + 2³ + 3³ + 4³ + 5³

Áp dụng công thức: 1³ + 2³ + 3³ +...+ n³  = \(\left(\dfrac{n\left(n+1\right)}{2}\right)^2\)

ta có: 1³ + 2³ + 3³ + 4³ + 5³ = \(\left(\dfrac{5.\left(1+5\right)}{2}\right)^2\) = 15² = 225

= 

225

1³ + 2³ + 3³  + 4³  + 5 ³

=1+8+27+64+125

= 225

a, A =  2 10 - 2 5 = 1024  - 32 = 992

b, B =  4 3 - 4 2 - 4 = 64  -  16  -  4 = 44

c, C =  3 2 . 2 3 + 4 3 . 2 5 = 9.8 + 64.32 = 2120

d, D =  1 3 + 2 3 + 3 3 + 4 3 + 5 3 = 1 + 8 + 27 + 64 + 125 = 225

a) A   =   2 10   -   2 5   =   1024   -   32   =   992 .

b) B   =   4 3   -   4 2   -   4   =   64   -   16   -   4   =   44 .

c) C   =   3 2   . 2 3   + 4 3 . 2 5   =   9 . 8   +   64 . 32   =   2120 .

d)  D   =   1 3   +   2 3   +   3 3   +   4 3   +   5 3   =   1   +   8   +   27   +   64   +   125   =   225