Question

Sa se arate ca punctele An(n+5,2n-3) ,n aparține lui N* sunt coloniare

Answer 1

$A_n(n+5,2n-3),\quad n\in \mathbb{N}^* \\ \\ d:~~y = ax+b\\ \\ A_n(n+5,2n-3)\in d \Rightarrow 2n-3 = a(n+5)+b,\quad \forall n\in \mathbb{N}\\\\ \Rightarrow 2n-3 = an+5a+b\\ \Rightarrow a = 2,\quad 5a+b = -3 \Rightarrow 10+b = -3 \Rightarrow b = -13\\ \\ \Rightarrow y = 2x-13\\ \\ A_n(n+5,2n-3) \in d\Rightarrow 2n-3 = 2(n+5)-13 \Rightarrow\\ \Rightarrow 2n-3 = 2n-3,\quad (A)\\ \\ \Rightarrow A_n(n+5,2n-3) \in y = 2x-13,\quad\forall~n\in \mathbb{R}$

$\Rightarrow A_n(n+5,2n-3)\in y = 2x-13,\quad\forall~n\in \mathbb{N}^*\\ \\\Rightarrow \text{Punctele }A_n(n+5,2n-3) \text{ sunt coliniare } \forall~n\in \mathbb{N}^*$