Question

Compară 2^3×2^6×2^9...×2^90 cu 8^450

Answer 1

   
[tex]\displaystyle\\ \text{De ls 3 la 90 avem: } ~ \frac{90-3}{3}+1 =\frac{87}{3}+1 =29+1 =30~\text{de termeni.} \\ \\ 2^3\times 2^6\times 2^9\times \cdots \times 2^{90}=2^{3+6+9+\cdots+90}= 2^{ \frac{30\times 91}{2} }=2^{15\times 91}=\boxed{2^{1365}} \\ \\ 8^{450} = \Big(2^3\Big)^{450} =2^{3\times 450}=\boxed{2^{1350}} \\ \\ 2^{1365} \ \textgreater \ 2^{1350} \\ \\ \Longrightarrow ~~~\boxed{\bold{2^3\times 2^6\times 2^9\times \cdots \times 2^{90} \ \textgreater \ 8^{450} }} [/tex]

Answer 2

2^ (3+6+...+90)com cu (2³)^450

3+6+...+90 = 3(1+3+...+30)=3*30*31/2=45*31
2^(45*31)comparat cu 2^(3*450)
2^(45*31)comparat cu 2^(3*45*10)
2^(45*31)comparat cu 2^(30*45)
2=2
45*31>45*30
primul numar e mai mare