`a, x - 75 = -26`

`x = -26 + 75` 

`x = 49`

Vậy/So...

\(b,\left(8-x\right).\left(x+15\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}8-x=0\\x+15=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=8-0\\x=0-15\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=8\\x=-15\end{matrix}\right.\)

Vậy \(x\in\left\{8;-15\right\}\)

\(A=3+3^2+3^3+...+3^{2024}\)

\(3A=3^2+3^3+3^4+...+3^{2025}\)

\(3A-A=\left(3^2+3^3+3^4+...+3^{2025}\right)-\left(3+3^2+3^3+...+3^{2024}\right)\)

\(2A=3^{2025}-3\)

\(A=\dfrac{3^{2025}-3}{2}\)

\(_{\left(\dfrac{13}{14}+\dfrac{1}{16}\right)-\left(\dfrac{13}{14}-\dfrac{15}{16}\right)}\)

\(=\dfrac{13}{14}+\dfrac{1}{16}-\dfrac{13}{14}+\dfrac{15}{16}\)

\(=\left(\dfrac{13}{14}-\dfrac{13}{14}\right)+\left(\dfrac{1}{16}+\dfrac{15}{16}\right)\)

\(=0+1=1\)

\(2\left(x-5\right)+3\left(x-9\right)=x-49\)

\(\Rightarrow2x-10+3x-27=x-49\)

\(\Rightarrow2x+3x-x=-49+27+10\)

\(\Rightarrow4x=-12\)

\(\Rightarrow x=-12:4\)

\(\Rightarrow x=-3\)

Vậy...

\(\left[461+\left(-78\right)+40\right]+\left(-461\right)\)

\(=\left[461-78+40\right]-461\)

\(=461-78+40-461\)

\(=\left(461-461\right)-\left(78-40\right)\)

\(=0-38=-38\)

`d, 125 . (-36) + (-36) . (-52) + 36 . (-27) + (-400)`

`= 125 . (-36) + (-36) . (-52) + (-36) . 27 + (-400)`

`= [125 + (-52) + 27] . (-36) + (-400)`

`= 100 . (-36) + (-400)`

`= (-3600) + (-400) = (-4000)`

\(1+5+5^2+...+5^{2024}\)

a,

Đặt \("1+5+5^2+...+5^{2024}"\) là `S`

Ta có :

\(S=1+5+5^2+...+5^{2024}\)

\(5S=5+5^2+5^3+...+5^{2025}\)

\(5S-S=\left(5+5^2+5^3+...+5^{2024}\right)-\left(1+5+5^2+...+5^{2024}\right)\)

\(4S=5^{2024}-1\)

\(S=\dfrac{5^{2024}-1}{4}\)

`b,`

\(4\times\dfrac{5^{2024}-1}{4}+1=5^n\)

\(\Rightarrow5^{2024}-1+1=5^n\)

\(\Rightarrow5^{2024}=5^n\)

\(\Rightarrow n=2024\)

\(2\left(x-5\right)-3\left(x+7\right)=14\)

\(\Rightarrow2x-10-3x+21=14\)

\(\Rightarrow2x-3x=14+21+10\)

\(\Rightarrow-x=45\)

\(\Rightarrow x=-45\)

Vậy...

\(B=2+2^2+2^3+...+2^{30}\)

\(B=\left(2+2^2+2^3+2^4+2^5+2^6\right)+...+\left(2^{25}+2^{26}+2^{27}+2^{28}+2^{29}+2^{30}\right)⋮21\)\(B=2.\left(1+2+2^2+2^3+2^4+2^5\right)+...+2^{25}.\left(1+2+2^2+2^3+2^4+2^5\right)⋮21\)\(B=2.\left(1+2+4+8+16+32\right)+2^{25}.\left(1+2+4+8+16+32\right)⋮21\)

\(B=\left(2+...+2^{25}\right).63⋮21\)

\(B⋮21\)

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

Ta có:

\(A=\left(5+5^2\right)+\left(5^3+5^4\right)+...+\left(5^{39}+5^{40}\right)⋮2\)

\(A=5.\left(1+5\right)+5^3.\left(1+5\right)+...+5^{39}.\left(1+5\right)⋮2\)

\(A=\left(1+5\right).\left(5+5^3+...+5^{39}\right)⋮2\)

\(A=6.\left(5+5^3+...+5^{39}\right)⋮2\)

\(A⋮2\)

Ta có:

\(A=\left(5+5^2+5^3+5^4\right)+...+\left(5^{37}+5^{38}+5^{39}+5^{40}\right)⋮3\)

\(A=5.\left(1+5+5^2+5^3\right)+...+5^{37}.\left(1+5+5^2+5^3\right)⋮3\)

\(A=\left(1+5+25+125\right).\left(5+...+5^{37}\right)⋮3\)

\(A=156.\left(5+...+5^{37}\right)⋮3\)

\(A⋮3\)