Question

a) aduceti la o forma mai simpla numarul a= 17+ 16 . 17 + 16 . 17 la a 2 +16 . 17 la a 3 + ............. + 16 . 17 la 2003
b) scrieti numarul a ca o suma de 17 numere naturale consecutive

Answer 1

$a)~Observam ~ca~17^k+16 \cdot 17^k=17^k(1+16)=17^k \cdot 17=17^{k+1}.\\ \\ A=(17+16 \cdot 17)+16 \cdot 17^2+16 \cdot 17^3+...+16 \cdot 17^{2003}= \\ \\ =(17^2+16 \cdot 17^2)+16 \cdot 17^3+...+16 \cdot 17^{2003}= \\ \\ =(17^3+16 \cdot 17^3)+...+16 \cdot 17^{2003}= \\ \\ =.................................= \\ \\ =17^{2004}.$

$b)~A=17^{2004}= \\ \\ =(17^{2003}-8)+(17^{2003}-7)+(17^{2003}-6)+...+(17^{2003}-1)+ \\ \\ +17^{2003}+(17^{2003}+1)+...+(17^{2003}+6)+(17^{2003}+7)+(17^{2003}+8)$