Đặt \(A=1+5^2+5^4+...+5^{40}\)

\(\Rightarrow25A=5^2+5^4+5^6+...+5^{42}\)

Lấy \(25A-A=\left(5^2+5^4+5^6+...+5^{42}\right)-\left(1+5^2+5^4+...+5^{40}\right)\)

\(\Rightarrow24A=5^{42}-1\)

\(\Rightarrow A=\dfrac{5^{42}-1}{24}\)

nguyễn thị hương giang, cúm ơn rất nhìu !!

Lời giải:
a. Ta thấy:

$3+3^2+3^3+...+3^{99}\vdots 3$

$1\not\vdots 3$

$\Rightarrow A=1+3+3^2+...+3^{99}\not\vdots 3$

$\Rightarrow A\not\vdots 9$

b.

$A=(5+5^2)+(5^3+5^4)+...+(5^{39}+5^{40})$

$=5(1+5)+5^3(1+5)+...+5^{39}(1+5)$

$=5.6+5^3.6+....+5^{39}.6$

$=6(5+5^3+...+5^{39})$

$=2.3.(5+5^3+...+5^{39})$

$\Rightarrow A\vdots 2$ và $A\vdots 3$

a) 11070 : {15 . [ 356 – ( 2110 – 2000 )]}

= 11070 : [15(356 – 110)] = 11070 : 3690 = 3

b) 62500 : { 50 2  : [ 112 – ( 52 –  2 3 . 5 )]}

= 62500 : { 2500 : [ 112 – ( 52 – 40 )]}

= 62500 : { 2500 : [ 112 – 12 ]}

= 62500 : { 2500 : 100 }

= 62500 : 25

= 2500

c)  3 3  .  5 3  – 20 . { 300 – [ 540 –  2 3  (  7 8  :  7 6  +  7 0  )]}

=  3 3  .  5 3  – 20 . {300 – [ 540 –  2 3 (72 + 1 )]

=  3 3  .  5 3  – 20 . [ 300 – (540 - 8 . 50)

= 27 . 125 – 20 . [300 – ( 540 - 400 )]

= 3375 – 20 . ( 300 – 140 )

= 3375 – 20 . 160

= 3375 – 3200

= 175

`5x-52=10`

`5x=10+52`

`5x=62`

`x=62:5`

`x=62/5=12,4`

`5x-52=10`

`5x=10+52`

`5x=62`

`x=62/5`

\(125:5^x+5^2=26\)

\(\Rightarrow125:5^x+25=26\)

\(\Rightarrow125:5^x=26-25\)

\(\Rightarrow125:5^x=1\)

\(\Rightarrow5^x=125:1\)

\(\Rightarrow5^x=125\)

\(\Rightarrow5^x=5^3\)

\(\Rightarrow x=3\)

a) ( 725 + 275 ) : 2 = 1000:2

= 500

b) 1458 x ( 460 + 540 ) = 1458 x 1000

= 1458000

d) (120 + 112 ) + ( 116 + 84 ) = 232 + 200

= 432 

725:2+275:2=275

1458x460+1458x540=1458000

52+54+55+56+...+88=2536

120+116+112+84=432

5*(5*x+x:5)=52*5

5*5*x+x:5*5=260

25*x+x=260

26*x=260

x=260*26

x=10

bạn chỉ mình phép trên được không, phép dưới mình viết sai

5x+x:5=52

=>5.5.x+x:5.5=260

=25x+x=260

26x=260

=>x=10

\(3\cdot(-2)^{5x+8}+5^2=1\\\Rightarrow 3\cdot(-2)^{5x+8}+25=1\\\Rightarrow 3\cdot(-2)^{5x+8}=1-25\\\Rightarrow 3\cdot(-2)^{5x+8}=-24\\\Rightarrow (-2)^{5x+8}=-24:3\\\Rightarrow (-2)^{5x+8}=-8\\\Rightarrow (-2)^{5x+8}=(-2)^3\\\Rightarrow 5x+8=3\\\Rightarrow 5x=3-8\\\Rightarrow 5x=-5\\\Rightarrow x=-5:5\\\Rightarrow x=-1\)

\(5x+2x\cdot\left(2^3\cdot5-3^2\cdot4\right)+5^2=4^3\)

\(\Rightarrow5x+2x\cdot\left(8\cdot5-9\cdot4\right)+25=64\)

\(\Rightarrow5x+2x\cdot\left(40-36\right)=64-25\)

\(\Rightarrow5x+2x\cdot4=39\)

\(\Rightarrow5x+8x=39\)

\(\Rightarrow x\cdot\left(5+8\right)=39\)

\(\Rightarrow13x=39\)

\(\Rightarrow x=\dfrac{39}{13}\)

\(\Rightarrow x=3\)

Vậy: ...