Question

Trebuie rezolvat cu P(h) si P(h+1)

Answer 1

$P(n):\quad 2\cdot 4^{2n+1}+5\cdot 3^{n+3}\,\,\vdots\,\,13,\quad \forall n\in \mathbb{N}^*\\ \\\\ P(k):\quad 2\cdot 4^{2k+1}+5\cdot 3^{k+3}\,\,\vdots\,\, 13\,\,\Leftrightarrow\,\,2\cdot 4^{2k+1}+5\cdot 3^{k+3} = M_{13}\\\\P(k+1):\quad 2\cdot 4^{2(k+1)+1}+5\cdot 3^{k+1+3} = \\ \\ =2\cdot 4^{2k+1+2}+5\cdot 3^{k+3+1} \\ \\ =2\cdot 4^{2k+1}\cdot 4^2+5\cdot 3^{k+3}\cdot 3\\ \\ = 2\cdot 4^{2k+1}\cdot 16+5\cdot 3^{k+3}\cdot 3$

$= 2\cdot 4^{2k+1}\cdot (13+3)+5\cdot 3^{k+3}\cdot 3\\ \\ =2\cdot 4^{2k+1}\cdot 13+2\cdot 4^{2k+1}\cdot 3 +5\cdot 3^{k+3}\cdot 3\\ \\ = M_{13}+3\cdot (2\cdot 4^{2k+1}+5\cdot 3^{k+3})\\ \\ = M_{13}+3\cdot M_{13}\\ \\ = M_{13}\quad \checkmark \\ \\\\\Rightarrow P(n)\to \text{adevarata}$