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Question Number 166379 by SANOGO last updated on 19/Feb/22

$$soit\{{}_{u_1=4}^{u_o=1}\mathrm{\forall }nϵN$$$$montrerparreccurenceque:$$$$U_{n+2}=2U_{n+1}-U_n$$

Answered by qaz last updated on 19/Feb/22

$${\mathrm{U}}_{\mathrm{n}+2}={2\mathrm{U}}_{\mathrm{n}+1}-{\mathrm{U}}_{\mathrm{n}}$$$$\Rightarrow {\lambda }^2-2\lambda +1=0$$$$\Rightarrow {\lambda }_1={\lambda }_2=1$$$$\Rightarrow {\mathrm{U}}_{\mathrm{n}}={\mathrm{C}}_1+{\mathrm{nC}}_2$$$${\mathrm{U}}_0=1\mathrm{\Lambda }{\mathrm{U}}_1=4$$$$\Rightarrow {\mathrm{U}}_{\mathrm{n}}=1+3\mathrm{n}$$

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