Question

Aratati ca D=7+7^2+7^3+...+7^2004 se divide cu 57

Answer 1

Răspuns

Grupam termenii convenabil.

Explicație pas cu pas:

$D=7+7^2+7^3+\ldots+7^{2004}\\D=(7+7^2+7^3)+(7^4+7^5+7^6)+\ldots + (7^{2002}+7^{2003}+7^{2004})\\D=7(1+7+7^2)+7^4(1+7+7^2)+\ldots + 7^{2002}(1+7+7^2)\\D=(1+7+7^2)(7+7^4+\ldots + 7^{2002})\\D=57 (7+7^4+\ldots +7^{2002})\\D\vdots 57$