Question

AJUTORR!! va roggg....

Answer 1

$daca~un~numar~natural~este~cuprins~intre~doua~patrate~perfecte~\\consecutive,~atunci~numarul~nu~este~un~patrat~perfect.\\------------------------\\a=\sqrt{\sum_{i=1}^n 2i}=\sqrt{2\sum_{i=1}^ni} =\sqrt{2\cdot\frac{n(n+1)}{2}}=\sqrt{n(n+1)}.\\a \in \mathbb{N} \Leftrightarrow n(n+1)~este~patrat~perfect.\\ne~propunem~deci~sa~demonstram~ca~n(n+1)~nu~este~un~patrat~perfect.\\n^2\leq n(n+1) \leq (n+1)^2 \Leftrightarrow n^2\leq n^2+n\leq n^2+2n+1\bigg|-n^2 \implies\\0\left\leq n\leq 2n+1,~evident.$

$prin~urmare~cum~n(n+1)~nu~este~patrat~perfect,~a~nu~este~un~numar\\natural.\\\\\Big(daca~n=0,~atunci~n(n+1)=0=0^2,~dar~problema~ne~spune~ca~n\in\mathbb{N}^* \Big)$