\(21.[(1 245 + 987) : 2^3 - 15.12] + 21\)

= 21.(2232 : 8 - 180) + 21

= 21.(279 - 180) + 21

= 21. 99 + 21

= 21. (99+1)

= 21. 100

= 2 100.

\(A=2+2^2+2^3+2^4+...+2^{99}+2^{100}\)

\(\Rightarrow2A=2^2+2^3+2^4+...+2^{100}+2^{101}\)

\(\Rightarrow A=2A-A=2^2+2^3+2^4+...+2^{100}+2^{101}-2-2^2-2^3-2^4-...-2^{99}-2^{100}=2^{101}-2\)

1/

Tổng A là tổng các số hạng cách đều nhau 4 đơn vị.

Số số hạng: $(101-1):4+1=26$

$A=(101+1)\times 26:2=1326$

2/

$B=(1+2+2^2)+(2^3+2^4+2^5)+(2^6+2^7+2^8)+(2^9+2^{10}+2^{11})$

$=(1+2+2^2)+2^3(1+2+2^2)+2^6(1+2+2^2)+2^9(1+2+2^2)$

$=(1+2+2^2)(1+2^3+2^6+2^9)$

$=7(1+2^3+2^6+2^9)\vdots 7$

3/
$C=1+2+2^2+2^3+...+2^{99}$

$2C=2+2^2+2^3+2^4+...+2^{100}$

$\Rightarrow 2C-C=2^{100}-1$

$\Rightarrow C=2^{100}-1$

\(23\cdot\left(25\cdot127-25\cdot87\right)+87\cdot25\cdot23-23\cdot25\cdot126\)

\(=23\cdot25\left(127-87\right)+23\cdot25\cdot87-23\cdot25\cdot126\)

\(=575\cdot40+575\cdot87-575\cdot126\)

\(=575\left(40+87-126\right)\)

\(=575\cdot1=575\)

23(25.127 - 25.87) + 87.25.23 - 23.25.126

= 23.25.127 - 23.25.87 + 87.25.23 - 23.25.126

= 23. 25.127 + 0 - 23.25.126

=23.25(127-126)

= 23. 25

= 575

23x(25x127-25x87)+87x25x23-23x25x126

= 23x25(127-87)+23x25(87-126)

=23x25(127-87+87-126)

=23x25=575

a) ( 71 + x ) - ( -24 - x ) + ( -35 - x )

= 71 + x + 24 + x - 35 - x

= ( 71 + 24 - 35 ) + ( x + x - x )

= 60 + x

b) x - 34 - [( 15 + x ) - ( 23 - x )]

= x - 34 - ( 15 + x - 23 + x )

= x - 34 - ( 15 - 23 + x2 )

= x - 34 - ( - 8 + 2x )

= x - 34 + 8 - 2x

= x - 2x + 8 - 34

= -x - 26

c) ( -15 + |x| ) + ( -25 - |x| )

= -15 + |x| - 25 + |x|

= -15 - 25 + x + x

= -40 + x2

a)(71+x)-(-24-x)+(-35-x)

=>71+x+24+x-35-x

=>60+x

b)x-34-[(15+x)-(23-x)]

=>x-34-(15+x-23+x)

=>x-34-15-x+23-x

=-x-26

71 + ( 26 - 3 . x ) ; 5 = 75 

        ( 26 - 3x ) : 5 = 75 - 71 

         ( 26 - 3x ) : 5 = 4 

          26 - 3x = 4.5 

          26 - 3x = 20

               3x = 26 - 20

               3x = 6

                x = 6 : 3 

                x = 2

\(A=2+2^3+...+2^{101}\)

\(4A=2^3+2^5+...+2^{101}+2^{103}\)

\(4A-A=2^{103}-2\)

\(3A=2^{103}-2\)

\(A=\dfrac{2^{103}-2}{3}\)

\(\Rightarrow1+2+2^3+...+2^{101}=A+1=\dfrac{2^{103}+1}{3}\)

giúp minh

chứng minh đúng ko?

\(2^2+2^3+2^4+2^5+...+2^{99}=2^2\left(1+2\right)+2^4\left(1+2\right)+...+2^{98}\left(1+2\right)=3.2^2+3.2^4+...+3.2^{98}=3\left(2^2+2^4+...+2^{98}\right)⋮3\)

\(B=2^2+2^3+...+2^{99}\)

\(B=\left(2^2+2^3\right)+...+\left(2^4+2^5\right)+...+\left(2^{98}+2^{99}\right)\)

\(B=3.2^2+3.2^4+...+3.2^{98}\)

\(B=3.\left(2^2+2^4+...+2^{98}\right)\)

\(\Rightarrow B⋮3\)