Question

Exercitiul din imagine.

Answer 1

$\displaystyle Observam~ca~\underbrace{111...1}_\mbox{k}}= \frac{10^k-1}{9} . \\ \\ Si~notand~t=10^{n-1},~avem: \\ \\ a= \frac{100t-1}{9} \cdot \frac{t-1}{9}+t= \frac{100t^2-20t+1}{81}= \\ \\ = \Big(\frac{10t-1}{9} \Big)^2 = \Big( \frac{10^n-1}{9} \Big)^2=\underbrace{111...1}_ \mbox{n}}^{~~~~~~~~~~2} =~patrat~perfect.$