a, \(M=5+5^2+5^3+...+5^{100}\)

\(\Rightarrow5M=5^2+5^3+5^4+...+5^{101}\)

\(\Rightarrow5M-M=\left(5^2+5^3+5^4+...+5^{101}\right)-\left(5+5^2+5^3+....+5^{100}\right)\)

\(\Rightarrow4M=5^{101}-5\)

\(\Rightarrow M=\frac{5^{101}-5}{4}\)

Vậy : \(M=\frac{5^{101}-5}{4}\)

bằng ?

b, \(N=5^1+5^2+5^3+...+5^{2010}\)

\(N=\left(5^1+5^2\right)+\left(5^3+5^4\right)+...+\left(5^{2009}+5^{2010}\right)\)

\(N=5\left(1+5\right)+5^3\left(1+5\right)+...+5^{2009}\left(5+1\right)\)

\(N=5.6+5^3.6+...+5^{2009}.6\)

\(N=\left(5+5^3+...+5^{2009}\right).6\)

\(\Rightarrow N⋮6\)

------------

\(N=5^1+5^2+5^3+5^4+...+5^{2010}\)

\(N=\left(5^1+5^2+5^3\right)+\left(5^4+5^5+5^6\right)+...+\left(5^{2008}+5^{2009}+5^{2010}\right)\)

\(N=5\left(1+30\right)+5^4\left(1+30\right)+...+5^{2008}\left(1+30\right)\)

\(N=5.31+5^4.31+...+5^{2008}.31\)

\(N=\left(5+5^4+...+5^{2008}\right).31\)

\(\Rightarrow N⋮31\)

A=2^1+2^2+2^3+2^4+...+2^2010 

=(2+2^2)+(2^3+2^4)+...+(2^2010+2^2011)

=2.(1+2)+2^3.(1+2)+...+2^2010.(1+2)

=2.3+2^3.3+...+2^2010.3

=(2+2^3+2^2010).3

=> A chia het cho 3

​​​​ 

Mà câu c bạn đánh chia hết thành chết hết rồi kìa

em chịu!!!!!!!!!!!

a) \(M=5+5^2+5^3+...+5^{100}\)

=> \(5M=\left(5+5^2+5^3+...+5^{100}\right).5\)

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

=> \(5M-M=\left(5^2+5^3+5^4+...+5^{101}\right)-\left(5+5^2+5^3+...+5^{100}\right)\)

=> \(4M=5^{101}-5\)

=> \(M=\frac{5^{101}-5}{4}\)

 ( 2¹ + 2² ) + ( 2³ + 2⁴ ) + ... + ( 2²⁰⁰⁹ + 2²⁰¹⁰ )

= 2. ( 1 + 2 ) + 2³ . ( 1 + 2 ) + ... + 2²⁰⁰⁹ . ( 1 + 2 )

= 3 . ( 2 + 2³ + ... + 2²⁰⁰⁹ ) chia hết cho 3. => ĐPCM

Minh lam cau A) thoi duoc hong

Câu 3: 

a: \(\Leftrightarrow n-1+4⋮n-1\)

\(\Leftrightarrow n-1\in\left\{1;-1;2;-2;4;-4\right\}\)

hay \(n\in\left\{2;0;3;-1;5;-3\right\}\)

b: \(\Leftrightarrow4n+2+1⋮2n+1\)

\(\Leftrightarrow2n+1\in\left\{1;-1\right\}\)

hay \(n\in\left\{0;-1\right\}\)

c: \(\Leftrightarrow4n-5=13k\left(k\in Z\right)\)

\(\Leftrightarrow n=\dfrac{13k+5}{4}\)

Câu b, chuyển 3^2010 thành 2^2010 nhé!

\(A=5+5^2+5^3+5^4+........+5^{2010}\)

A = ( 1 + 5 + 5² ) + ............ + ( 5²⁰⁰⁸ + 5²⁰⁰⁹ + 5²⁰¹⁰ )

A = 31 + ......... + 31( 1 + 5 + 5² )

Mà 31\(⋮\)31 => A \(⋮\)31 ( đpcm )

đề bài sai rồi

ko sai đâu bạn ạ