Question

cho A = 3^1 + 3^2 +3^3 +3^4+...+3^2012.chứng minh rằng A chia hết cho 120

Answer 1

\(A=\left(3+3^2+3^3+3^4\right)+3^4\left(3+3^2+3^3+3^4\right)+...+3^{2008}\left(3+3^2+3^3+3^4\right)\)

\(=120+3^4.120+...+3^{2008}.120=120\left(1+3^4+...+3^{2008}\right)⋮120\)

Answer 2

\(A=\left(3+3^2+3^3+3^4\right)+...+\left(3^{2009}+3^{2010}+3^{2011}+3^{2012}\right)\)

\(A=\left(3+3^2+3^3+3^4\right)+...+3^{2008}\left(3+3^2+3^3+3^4\right)\)

\(A=\left(3+3^2+3^3+3^4\right)\left(1+3^4+...+3^{2008}\right)\)

\(A=120\left(1+3^4+...+3^{2008}\right)⋮120\)

Answer 3

\(A=3+3^2+3^3+...+3^{2012}\)

\(A=\left(3+3^2+3^3+3^4\right)+\left(3^5+3^6+3^7+3^8\right)+...+\left(3^{2009}+...+3^{2012}\right)\)

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

\(A=3.40+3^5.40+...+3^{2009}.40\)

\(A=120+3^4.120+...+3^{2008}.120\)

\(A=120\left(1+3^4+...+3^{2008}\right)⋮120\)

Answer 4

Ta có :

\(A=3^1+3^2+3^3+...+3^{2012}\)

\(=\left(3^1+3^2+3^3+3^4\right)+\left(3^5+3^6+3^7+3^8\right)+...+\left(3^{2009}+3^{2010}+3^{2011}+3^{2012}\right)\)

\(=3\left(1+3+3^2+3^3\right)+...+3^{2009}\left(1+3+3^2+3^3\right)\)

\(=3.40+...+3^{2009}40\)

\(=120+...+3^{2008}\left(3.40\right)\)

\(=120+...+3^{2008}.120\)

\(120⋮̸120\)

\(\Leftrightarrow A⋮120̸\)(đpcm)

Answer 5

A=(3+3^2+3^3+3^4)+3^4*(3+3^2+3^3+3^4)+3^2008*(3+3^2+3^3+3^4)

=120+3^4*120+...+2^2008*120

=120*(1+3^4+...+3^2008)chia hết cho 120