Question

CMR:
a/\(55^{n+1}-55n\) chia hết cho 54 với mọi\(x\in N\)

Ta có \(55^{n+1}-55^n=......................\)

b/\(n^2\left(n+1\right)+2n\left(n+1\right)\) luôn chia hết cho 6 với mọi số nguyên n.
Ta có:\(n^2\left(n+1\right)+2n\left(n+2\right)=.......\)

c/\(2^{n+2}+2^{n+1}+2^n\) chia hết cho 7,với mọi\(x\in N\).

Ta có:\(2^{n+2}+2^{n+1}+2^n=...\)

Answer 1

a)

\(55^{n+1}-55^n\\ =55^n.55-55^n\\ =55^n\left(55-1\right)\\ =55^n.54⋮54\\ \RightarrowĐpcm\)

b)

\(n^2\left(n+1\right)+2n\left(n+1\right)\\ =\left(n+1\right)\left(n^2+2n\right)\\ =n\left(n+1\right)\left(n+2\right)⋮6\\ \)

c)

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