Question

CM đẳng thức: 1+5+5²+...+5²⁰¹²=5^{2013-1 / 4}

^{dấu chia trên là phân số}

 

Answer 1

Ta có : \(1+5+5^2+...+5^{2012}\)

Đặt : \(A=1+5+5^2+...+5^{2012}\)

\(\Rightarrow5A=5+5^2+5^3+...+5^{2013}\)

\(\Rightarrow5A-A=\left(5+5^2+...+5^{2013}\right)-\left(1+5+...+5^{2012}\right)\)

\(\Rightarrow4A=5^{2013}-1\)( Trừ vế theo vế )

\(\Rightarrow A=\frac{5^{2013}-1}{4}\left(đpcm\right)\)

Answer 2

Đặt \(A=1+5+5^2+...+5^{2012}\)

Ta có : \(5A=5+5^2+5^3+...+5^{2013}\)

\(\Rightarrow5A-A=\left(5+5^2+5^3+...+5^{2013}\right)-\left(1+5+5^2+...+5^{2012}\right)\)

\(\Rightarrow4A=5^{2013}-1\Rightarrow A=\frac{5^{2013}-1}{4}\RightarrowĐPCM\)

Answer 3

Đặt \(VT=A=1+5+5^2+......+5^{2012}\)

\(\Rightarrow5A=5+5^2+5^3+.....+5^{2013}\)

\(\Rightarrow5A-A=\left(5+5^2+5^3+....+5^{2013}\right)-\left(1+5+5^2+....+5^{2012}\right)\)

\(\Rightarrow4A=5^{2013}-1\)

\(\Rightarrow A=\frac{5^{2013}-1}{4}=VT\)

Vậy đẳng thức được chứng minh