mk đg cần gấp ạ. TKS mn

Ta thấy : `|x-1|+|x-5|=|x-1|+|5-x|>=|x-1+5-x|=4`

 `|x-3|+|x-7|=|x-3|+|7-x|>=|x-3+7-x|=4`

`->|x-1|+|x-3|+|x-5|+|x-7|>=4+4=8` ( đpcm )

a)

\(7^6+7^5-7^4=7^4\left(7^2+7-1\right)=7^4.55\) chia hết cho 55 (đpcm )

b)

\(16^5+2^{15}=\left(2^4\right)^5+2^{15}=2^{20}+2^{15}=2^{15}\left(2^5+1\right)=2^{15}.33\) chia hết cho 33 (đpcm )

c)

\(81^7-27^9-9^{13}=\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}=3^{28}-3^{27}-3^{26}\)

\(=3^{22}\left(3^6-3^5-3^4\right)=3^{22}.405\) chia hết cho 405 (đpcm )

-Schwarz: 1/(a+b)+1/(a+c)+1/(b+c) >/ 9/2(a+b+c)=9/2=4,5>4 -> đpcm 

-ta có VT=4(1-a)(1-b)(1-c)=4(b+c)(1-b)(1-c)=[4(b+c)(1-c)](1-b)

Áp dụng bdt cauchy dạng 4ab </ (a+b)^2 

VT </ (b+c+1-c)^2(1-b)=(b+1)^2(1-b)=(b+1)[(1+b)(1-b)]=(b+1)(1-b^2) </ 1+b = a+2b+c (đpcm) 

Ta có: \(A=7+7^2+7^3+7^4+...+7^{4n-3}+7^{4n-2}+7^{4n-1}+7^{4n}\)

          \(A=\left(7+7^2+7^3+7^4\right)+...+\left(7^{4n-3}+7^{4n-2}+7^{4n-1}+7^{4n}\right)\)

         \(A=7\left(1+7+7^2+7^3\right)+...+7^{4n-3}\left(1+7+7^2+7^3\right)\)

         \(A=7.400+7^5.400+...+7^{4n-3}.400\)

        \(A=400.\left(7+7^5+..+7^{4n-3}\right)\)luôn chia hết cho 400

A=7+7²+7⁴+7⁴+...+7^4n-3+7^4n-2+7^4n-1+7⁴ⁿ

          A=(7+7²+7³+7⁴)+...+(7^4n-3+7^4n-2+7^4n-1+7⁴ⁿ)

         A=7(1+7+7²+7³)+...+7^4n-3(1+7+7²+73)

         A=7.400+7⁵.400+...+7^4n-3.400

        A=400.(7+7⁵+..+7^4n-3)luôn chia hết cho 400

\(P=1+3+3^2+...+3^7\)

\(=\left(1+3\right)+...+\left(3^6+3^7\right)\)

\(=1\left(1+3\right)+...+3^6\left(1+3\right)\)

\(=1\cdot4+...+3^6\cdot4\)

\(=4\cdot\left(1+...+3^6\right)⋮4\)

Đpcm

p=1+3+3²+3³+3⁴+3⁵+3⁶+3⁷

p=(1+3)+(3²+3³)+(3⁴+3⁵)+(3⁶+3⁷)

p=4.1+(3².1+3².3)+(3⁴.1+3⁴.3)+(3⁶.1+3⁶.3)

p=4.1+3²(1+3)+3⁴(1+3)+3⁶(1+3)

p=4.1+3².4+3⁴.4+3⁶.4

p=4.(1+3²+3⁴+3⁶)

vay P chia het cho 4  

Áp dụng tính chất `|P|>=P,|P|>=-P`

`=>{(|x+5|>=x+5),(|x+1|>=-x-1):}`

`=>|x+5|+|x+1|>=x+5-x-1=4`

Mặt khác:`|x+3|>=0`

`=>|x+1|+|x+3|+|x+5|>=4(đpcm)`

Dấu "=" xảy ra khi `x=-3`