Question

RPEDE DAU COROANA E URGENT SUNT DISPERATA!!!!!9 SI 11

Answer 1

   
[tex]9)~\text{Comparati:} \\ \\ a)~2^{28} ~cu~ 5^{14} \\ 2^{28} = 2^{2\times 14} = (2^2)^{14}= 4^{14} \\ 4^{14} \ \textless \ 5^{14} \\ \Longrightarrow~~2^{28} \ \textgreater \ 5^{14} \\ \\ b) ~3^{51} cu 4^{34} \\ 3^{51} = 3^{3\times 17} = 27^{17} \\ 4^{34} = 4^{2\times 17} = 16^{17} \\ 27^{17} \ \textgreater \ 16^{17} \\ \Longrightarrow~~3^{51} \ \textgreater \ 4^{34} \\ \\ c)~2^{105}~cu~3^{63} \\ 2^{105} = 2^{5 \times 21} = 32^{21} \\ 3^{63}= 3^{3 \times 21} = 27^{21} \\ 32^{21} \ \textgreater \ 27^{21} \\ \Longrightarrow~~2^{105}\ \textgreater \ 3^{63} [/tex]

[tex]d)~4^{111}~cu~6^{74} \\ 4^{111} = 4^{3 \times 37} = 64^{37} \\ 6^74 = 6{2 \times 37} = 36^37 \\ 64^{37} \ \textgreater \ 36^37 \\ \Longrightarrow~~4^{111}\ \textgreater \ 6^{74}[/tex]


[tex]11) \\ a)~ S=2^{14}+2^{16}+2^{18} = 2^14(1 + 2^2+2^4)=2^{14}\times 21 \\ \Longrightarrow~~S~\vdots ~ 2^{14} ~si~S~\vdots ~ 3~si~S~\vdots ~ 7 \\ \\ b) ~S=3^{15}+3^{16}+3^{17} = 3^{15}(1 + 3 + 3^2) =3^{15} \times 13 \\ \Longrightarrow~~S~\vdots ~ 3^{15} ~si~S~\vdots ~ 13 [/tex]


[tex]c)~S = 5^1+5^2+5^3+\hdots+5^{101}+5^{102} \\ Sirul are 102 termeni. \\ \text{Putem grupa sirul in grupe de cate 2 termeni }\\ \text{deoarece 102 este numar par.} \\ \text{Prima grupa este: } \\ 5^1+5^2 = 5+25 = 30\\   \text{Celelalte grupe sunt multiplii ai primei grupe.} \\ \Longrightarrow~~S~\vdots ~ 30 \\ \Longrightarrow~~S~\vdots ~ 2~si~ S~\vdots ~ 3~si~ S~\vdots ~ 5 [/tex]


[tex]\text{Putem grupa sirul in grupe de cate 3 termeni }\\ \text{deoarece 102 este multiplu de 3.} \\ \text{Prima grupa este: } \\ 5^1+5^2+5^3 = 5+25 +125= 155 \\ \text{Celelalte grupe sunt multiplii ai primei grupe.} \\ \Longrightarrow~~S~\vdots ~ 155 \\ \Longrightarrow~~S~\vdots ~ 5~si~ S~\vdots ~ 31 [/tex]