Question

Chứng minh rằng a = 5ⁿ⁺² +5ⁿ⁺¹ +5ⁿ chia hết cho 31

Answer 1

\(CM:a=5^{n+2}+5^{n+1}+5^n⋮31\)
\(a=5^{n+2}+5^{n+1}+5^n\)
=> \(a=5^n.5^2+5^n.5+5^n\)
=> \(a=5^n\left(5^2+5+1\right)\)
=> \(a=5^n.31\)
Vì \(31⋮31\)=> \(5^n.31⋮31\)
=> \(a⋮31\)(\(đpcm\))

Answer 2

a = 5\(^{n+2}\) + 5\(^{n+1}\)+5\(^n\)

= 5\(^n\) .5\(^2\) + 5\(^n\).5 + 5\(^n\)

= 5\(^n\) ( 5\(^2\) +5+1)

= 5\(^n\)(25+5+1) = 5\(^n\) .31 \(⋮\) 31

Answer 3

Ta có : \(a=5^{n+2}+5^{n+1}+5^n\)

\(\Rightarrow a=5^n.5^2+5^n.5+5^n\)

\(\Rightarrow a=5^n.\left(5^2+5+1\right)\)

\(\Rightarrow a=5^n.31\) \(⋮31\) (đpcm)