Question

2^{n} ·5^{n+1} se poate scrie ca o suma a doua patrate perfecte,oricare ar fi „n„ nr natural.

Answer 1

[tex]\text{Daca $n$ este par atunci $n=2k$}\\ 2^{2k}\cdot5^{2k+1}=2^n\cdot5^{2k}\cdot5=10^{2k}\cdot5=10^{2k}(1+4)=10^{2k}+10^{2k}\cdot2^2\\ =(10^k)^2+(2\cdot10^k)^2\\ \text{Daca $n$ este impar atunci $n=2k+1$}\\ 2^{2k+1}\cdot5^{2k+2}=2^{2k}\cdot5^{2k}\cdot2\cdot5^2=2^{2k}\cdot5^{2k}\cdot50=\\ 2^{2k}\cdot5^{2k}\cdot(1+49)=2^{2k}\cdot5^{2k}\cdot1+2^{2k}\cdot5^{2k}\cdot49=\\ =(10^k)^2+(7\cdot10^k)^2 [/tex]