`#3107`

b)

`B = 2 + 2^2 + 2^3 + ... + 2^9 + 2^10`

`= (2 + 2^2 + 2^3 + 2^4 + 2^5) + (2^6 + 2^7 + 2^8 + 2^9 + 2^10)`

`= 2.(1 + 2 + 2^2 + 2^3 + 2^4) + 2^6 . (1 + 2 + 2^2 + 2^3 + 2^4)`

`= 2.(1 + 2 + 4 + 8 + 16) + 2^6 . (1 + 2 + 4 + 8 + 16)`

`= 2.31 + 2^6.31`

`= 31.(2 + 2^6)`

Vì `31 \vdots 31`

`\Rightarrow 31.(2 + 2^6) \vdots 31`

Vậy, `B \vdots 31 (đpcm).`

B = 2 + 2² + 2³ + ... + 2⁹ + 2¹⁰

B = 2(1 + 2¹ + ... + 2⁴) + 2⁵(1 + 2¹ + ... + 2⁴)

B = (2 + 2⁵).(1 + 2 + 4 + 8 + 16)

B = (2 + 2⁵).31

Do đó B chia hết cho 31

a)

A = 3 + 3² + 3³ + ... + 3⁶

3A = 3² + 3³ + 3⁴ + ...  + 3⁷

3A - A = (3² + 3³ + 3⁴ + ...  + 3⁷) - (3 + 3² + 3³ + ... + 3⁶)

2A = 3⁷ - 3

A = \(\dfrac{3^7-3}{2}\)

b)

Từ câu a) suy ra

2A - 3 = 3^x

3⁷ - 3 - 3 = 3^x (rõ ràng đề sai)

c)

A = 3 + 3² + 3³ + ... + 3⁶

A = 3(1 + 3¹) + 3³(1 + 3¹) + 3⁵(1 + 3¹)

A = (3 + 3³ + 3⁵).4

Do đó A ⋮ 4

Bạn đã học số nguyên chưa

700 . 25

= 100 . 7 . 25

= 100 . 175

= 17500

700.25

= 175.4.25

= 175.100

= 17500 

200:n=m(dư 13)

=> 200=m.n+13

=> 200-13=m.n

=> 187=m.n

=> 11.17= m.n

=> m= 11; n= 17 hoặc m=17; n=11

Mà n là số chia nên n>13 => n=17; m=11

1hh

Ta có:

180 = 2².3².5

2000 = 2⁴.5³

450 = 2.3².5²

2020 = 2².5.101

Do đó số ước của 180; 2000; 450; 2020 lần lượt là:
(2 + 1)(2 + 1)(1 + 1) = 18 (ước)

(4 + 1)(3 + 1) = 20 (ước)

(1 + 1)(2 + 1)(2 + 1) = 18 (ước)

(2 + 1)(1 + 1)(1 + 1) = 12 (ước)

\(12.42+12.35+12^2+12\)

\(=12.42+12.35+12.12+12.1\)

\(=12.\left(42+35+12+1\right)\)

\(=12.90\)

\(=1080\)

`#3107`

b)

`2.3^x = 162`

`\Rightarrow 3^x = 162 \div 2`

`\Rightarrow 3^x = 81`

`\Rightarrow 3^x = 3^4`

`\Rightarrow x = 4`

Vậy, `x = 4`

c)

`(2x - 15)^5 = (2 - 15)^3`

\(\Rightarrow \)`(2x - 15)^5 - (2x - 15)^3 = 0`

\(\Rightarrow \)`(2x - 15)^3 . [ (2x - 15)^2 - 1] = 0`

\(\Rightarrow\left[{}\begin{matrix}\left(2x-15\right)^3=0\\\left(2x-15\right)^2-1=0\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}2x-15=0\\\left(2x-15\right)^2=1\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}2x=15\\\left(2x-15\right)^2=\left(\pm1\right)^2\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=\dfrac{15}{2}\\2x-15=1\\2x-15=-1\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=\dfrac{15}{2}\\2x=16\\2x=-14\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=\dfrac{15}{2}\\x=8\\x=-7\end{matrix}\right.\)

Vậy, `x \in`\(\left\{-7;8;\dfrac{15}{2}\right\}.\)

`d)`

\(3^{x+2}-5.3^x=?\) Bạn ghi tiếp đề nhé!

`e)`

\(7\cdot4^{x-1}+4^{x-1}=23?\)

\(4^{x-1}\cdot\left(7+1\right)=23\\ \Rightarrow4^{x-1}\cdot8=23\\ \Rightarrow4^{x-1}=\dfrac{23}{8}\)

Bạn xem lại đề!

`f)`

\(2\cdot2^{2x}+4^3\cdot4^x=1056\)

\(\Rightarrow2\cdot2^{2x}+\left(2^2\right)^3\cdot\left(2^2\right)^x=1056\\ \Rightarrow2\cdot2^{2x}+2^6\cdot2^{2x}=1056\\ \Rightarrow2^{2x}\cdot\left(2+2^6\right)=1056\\ \Rightarrow2^{2x}\cdot66=1056\\ \Rightarrow2^{2x}=1056\div66\\ \Rightarrow2^{2x}=16\\ \Rightarrow2^{2x}=2^4\\ \Rightarrow2x=4\\ \Rightarrow x=2\)

Vậy, `x = 2`

_____

\(10 -{[(x \div 3+17) \div 10+3.2^4] \div 10}=5\)

\(\Rightarrow\left[\left(x\div3+17\right)\div10+48\right]\div10=10-5\)

\(\Rightarrow\left[\left(x\div3+17\right)\div10+48\right]\div10=5\)

\(\Rightarrow\left(x\div3+17\right)\div10+48=50\)

\(\Rightarrow\left(x\div3+17\right)\div10=2\)

\(\Rightarrow x\div3+17=20\)

\(\Rightarrow x\div3=3\\ \Rightarrow x=9\)

Vậy, `x = 9.`

Lời giải:

$2+4+6+8+10+...+2500=2(1+2+3+4+...+1250)$

$=2.1250(1250+1):2=1250(1250+1)=x(x+1)$

$\Rightarrow x=1250$ (điều kiện $x$ là stn)

`#3107`

`a)`

`(2x + 1)^2 = 36?`

\(\Rightarrow\left(2x+1\right)^2=\left(\pm6\right)^2\\\Rightarrow\left[{}\begin{matrix}2x+1=6\\2x+1=-6\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}2x=6-1\\2x=-6-1\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}2x=5\\2x=-7\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=\dfrac{5}{2}\\x=-\dfrac{7}{2}\end{matrix}\right. \)

Vậy, \(x\in\left\{-\dfrac{7}{2};\dfrac{5}{2}\right\}.\)

\(\left(2x+1\right)^3=36\) -> kiểm tra lại đề nhé bạn.