Question

exercitiul 3 rpd.55puncte

Answer 1

[tex]z=3^{0}+3^{1} +3^{2} +3^{3}+...+3^{89} [/tex]

[tex]z=(1+3)+(3^{2} +3^{3})+...+(3^{88}+3^{89}) [/tex]

[tex]z=4+3^{2}(1+3)+3^{4}(1+3)+...+3^{88}(1+3) [/tex]

$z=4+3^{2}*4+3^{4}*4+...+3^{88}*4$

$z=4(3^{2}+3^{4}+3^{6}+...+3^{88})$

=> z divizibil cu 4

Proprozitia este adevarata.


$t=3^{0}+3^{1} +3^{2} +3^{3}+...+3^{92}$

$t=(3^{0}+3^{1} +3^{2})+(3^{3}+3^{4} +3^{5})+...+(3^{90}+3^{91} +3^{92})$

$t=(1+3+9)+3^{3}(1+3+3^{2})+3^{6}(1+3+3^{2})+...+3^{90}(1+3+3^{2})$

$t=13+3^{3}*13+3^{6}*13+...+3^{90}*13$

$t=13(1+3^{3}+3^{6}+3^{9}+...+3^{90}$

=> t divizibil cu 13

Proprozitia este adevarata.