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Question Number 95014 by I want to learn more last updated on 22/May/20

$$\mathbf{If}{\mathbf{y}}_{\mathbf{n}+1}-{\mathbf{y}}_{\mathbf{n}}=6,\mathbf{and}{\mathbf{y}}_0=7$$$$\mathbf{Find}{\mathbf{y}}_{\mathbf{n}}$$

Commented by prakash jain last updated on 22/May/20

$$\mathrm{This}\mathrm{is}\mathrm{an}\mathrm{arithmetic}\mathrm{progression}:$$$$y_0=7$$$$y_{n+1}=y_n+6$$$$\mathrm{firzt}\mathrm{term}7$$$$\mathrm{common}\mathrm{difference}=6$$$$y_n=7+n6$$

Commented by I want to learn more last updated on 22/May/20

$$\mathrm{Sir},\mathrm{with}{\mathrm{y}}_{\mathrm{n}}=\mathrm{a}+(\mathrm{n}-1)\mathrm{d},\mathrm{i}\mathrm{did}\mathrm{not}\mathrm{get}\mathrm{it}.$$$${\mathrm{y}}_{\mathrm{n}}=7+(\mathrm{n}-1).6=7+6\mathrm{n}-6=6\mathrm{n}+1$$

Commented by I want to learn more last updated on 22/May/20

$$\mathrm{And}\mathrm{sir}$$$$\mathrm{for}{\mathrm{y}}_{\mathrm{n}-2}+{\mathrm{y}}_{\mathrm{n}-1}+{\mathrm{y}}_{\mathrm{n}}=0$$$$\mathrm{auxilliary}\mathrm{equation}\mathrm{is}:{\mathrm{x}}^2+\mathrm{x}+1=0$$$$$$$$\mathrm{What}\mathrm{of}$$$${\mathrm{y}}_{\mathrm{n}+2}+{\mathrm{y}}_{\mathrm{n}+3}+{\mathrm{y}}_{\mathrm{n}-1}+{\mathrm{y}}_{\mathrm{n}}=0$$$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{auxilliary}\mathrm{equation}$$

Commented by mr W last updated on 22/May/20

$$‘‘youwanttolearn{more}^{″}sir:$$$$trytolearnthingsoneafteranother!$$$$ifyouarestillatthelevelforAP,$$$$forgetatfirst‘‘auxilliary{equation}^{″}.$$$$backtoyouroriginalquestion:$$$$youshouldmakeclearwhatyoumean$$$$with‘‘n^{″}.ifyousay{y}_n=6n+1,but$$$$yougaveyourfirstterm{y}_0,then$$$$youarewrong,since{y}_0=6\times 0+1=1\ne 7.$$$$correctis{y}_n=6n+7.soneverjust$$$$learnaformula,butlearnwhatis$$$$behindit!$$

Commented by I want to learn more last updated on 22/May/20

$$\mathrm{Thanks}\mathrm{sir}.$$

Commented by I want to learn more last updated on 22/May/20

$$\mathrm{But}\mathrm{i}\mathrm{got}\mathrm{the}\mathrm{question}\mathrm{from}\mathrm{the}\mathrm{auxilliary}\mathrm{equations}$$

Commented by prakash jain last updated on 22/May/20

$$\mathrm{You}\mathrm{may}\mathrm{want}\mathrm{to}\mathrm{just}\mathrm{do}\mathrm{a}\mathrm{internet}$$$$\mathrm{search}\mathrm{on}‘‘\mathrm{recurrence}{\mathrm{relation}}^{″}.$$$$\mathrm{You}\mathrm{will}\mathrm{find}\mathrm{some}\mathrm{general}\mathrm{techinques}$$$$\mathrm{on}\mathrm{how}\mathrm{to}\mathrm{solve}\mathrm{recurrence}\mathrm{relation}.$$$$\mathrm{Sometime}\mathrm{you}\mathrm{have}\mathrm{to}\mathrm{do}\mathrm{a}\mathrm{variable}$$$$\mathrm{subtituion}\mathrm{to}\mathrm{convert}\mathrm{to}\mathrm{linear}$$$$\mathrm{recurrence}(\mathrm{ex}.\mathrm{Q}93173)$$

Commented by I want to learn more last updated on 22/May/20

$$\mathrm{I}\mathrm{can}\mathrm{express}\mathrm{negative}\mathrm{base}\mathrm{like}$$$${\mathrm{y}}_{\mathrm{n}-2}+{\mathrm{y}}_{\mathrm{n}-1}$$$$\mathrm{but}\mathrm{i}\mathrm{cannot}\mathrm{find}\mathrm{aucilliary}\mathrm{of}$$$${\mathrm{y}}_{\mathrm{n}+2}+{\mathrm{y}}_{\mathrm{n}+3}+...\mathrm{for}\mathrm{positive}\mathrm{base}\mathrm{and}\mathrm{negative}\mathrm{together}.$$

Commented by prakash jain last updated on 22/May/20

$$\mathrm{What}\mathrm{matter}\mathrm{is}\mathrm{a}\mathrm{difference}\mathrm{say}$$$$\mathrm{for}\mathrm{example}$$$$y_{n+2}=5y_n+3y_{n+1}$$$$\mathrm{Subtitue}n=n-2$$$$y_n=5y_{n-2}+3y_{n-1}$$

Commented by I want to learn more last updated on 22/May/20

$$\mathrm{I}\mathrm{appreciate}.\mathrm{now}\mathrm{i}\mathrm{have}\mathrm{an}\mathrm{idea}.$$$$\mathrm{Thanks}\mathrm{for}\mathrm{your}\mathrm{time}.$$

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