Question

Arătați ca suma a opt puteri consecutive ale numărului 2 este multiplu de 17

Answer 1

2^n + 2^(n+1) + 2^(n+2) + … + 2^(n+7) 
= 2^n * (1 + 2 + 4 + 8 + 16 + 32 + 64 + 128)
= 2^n * (1 + 2 + 4 + 8 + 16 + 32 + 64 + 128)
= 255 * 2^n
= 17 * 15 * 2^n

Answer 2

$2^n+2^{n+1}+2^{n+2}+...+2^{n+7}=2^n(1+2^1+2^2+2^3+...+2^7)=2^n[2(2^7-1)+1]=2^n*255=2^n*15*17 =\textgreater 17|2^n*15*17 =\textgreater 17|2^n+2^{n+1}+2^{n+2}+...+2^{n+7}$