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Question Number 75327 by TawaTawa last updated on 09/Dec/19

$$\mathrm{If}{\mathrm{T}}_{\mathrm{n}+1}=1+\frac{1}{2}{\mathrm{T}}_{\mathrm{n}}$$$$\mathrm{Find}\mathrm{a}\mathrm{formular}\mathrm{for}{\mathrm{T}}_{\mathrm{n}}\mathrm{in}\mathrm{terms}\mathrm{of}\mathrm{n}$$$$\mathrm{and}\mathrm{find}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{first}\mathrm{n}\mathrm{terms}$$

Commented by malwaan last updated on 11/Dec/19

$${\mathit{T}}_1=4$$$${\mathit{T}}_2=1+\frac{1}{2}\times 4=1+2=3$$$${\mathit{T}}_3=1+\frac{1}{2}\times 3=1+\frac{3}{2}=\frac{5}{2}$$$${\mathit{T}}_4=1+\frac{1}{2}\times \frac{5}{2}=1+\frac{5}{4}=\frac{9}{4}$$$${\mathit{T}}_5=1+\frac{9}{8}=\frac{17}{8}=\frac{2^4+1}{2^3}=\frac{2^{5-1}+1}{2^{5-2}}$$$$\Rightarrow {\mathit{T}}_{\mathit{n}}=\frac{2^{\mathit{n}-1}+1}{2^{\mathit{n}-2}}$$

Commented by mr W last updated on 10/Dec/19

$$pleasecheckthequestion:$$$$whengivingarecursivesequence,$$$$thefirstterm{T}_{1}mustbegivenin$$$$ordertodetermineallterms.$$

Commented by TawaTawa last updated on 10/Dec/19

$$\mathrm{Ohh},\mathrm{yes}\mathrm{sir},{\mathrm{T}}_1=4$$$$\mathrm{Please}\mathrm{help}\mathrm{me}\mathrm{edit}\mathrm{it}\mathrm{in}\mathrm{your}\mathrm{work}.\mathrm{Thanks}\mathrm{sir}$$

Answered by mr W last updated on 10/Dec/19

$$say{T}_n={Ap}^n+B$$$$T_{n+1}={Ap}^{n+1}+B$$$${Ap}^{n+1}+B=1+\frac{{Ap}^n}{2}+\frac{B}{2}$$$$\Rightarrow {Ap}^n(p-\frac{1}{2})+\frac{B}{2}=1$$$$\Rightarrow p-\frac{1}{2}=0\Rightarrow p=\frac{1}{2}$$$$\Rightarrow \frac{B}{2}=1\Rightarrow B=2$$$$\Rightarrow {\mathit{T}}_{\mathit{n}}=\frac{\mathit{A}}{2^{\mathit{n}}}+2$$$$if{T}_1=4,$$$$4=\frac{A}{2}+2$$$$\Rightarrow A=4$$$$\Rightarrow {\mathit{T}}_{\mathit{n}}=2+\frac{1}{2^{\mathit{n}-2}}$$$$\underset{k=1}{\sum ^n}{T}_k=2n+\frac{2(1-\frac{1}{2^n})}{1-\frac{1}{2}}=2(n+2)-\frac{1}{2^{n-2}}$$

Commented by TawaTawa last updated on 10/Dec/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{I}\mathrm{appreciate}\mathrm{your}\mathrm{time}$$

Commented by TawaTawa last updated on 10/Dec/19

$$\mathrm{I}\mathrm{really}\mathrm{appreciate}\mathrm{sir}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{more}.$$

Commented by TawaTawa last updated on 10/Dec/19

$$\mathrm{Sir},\mathrm{is}\mathrm{there}\mathrm{any}\mathrm{hint}\mathrm{on}\mathrm{how}\mathrm{we}\mathrm{assume}{\mathrm{T}}_{\mathrm{n}}={\mathrm{Ap}}^{\mathrm{n}}+\mathrm{B}.$$$$\mathrm{Or}\mathrm{it}\mathrm{is}\mathrm{general}\mathrm{for}\mathrm{such}\mathrm{questions}$$

Commented by TawaTawa last updated on 10/Dec/19

$$\mathrm{And}\mathrm{please}\mathrm{sir},\mathrm{why}\mathrm{is}\mathrm{p}-\frac{1}{2}=0\mathrm{and}\frac{\mathrm{B}}{2}=1$$

Commented by mind is power last updated on 10/Dec/19

$${\mathrm{T}}_{\mathrm{n}+1}={\mathrm{aT}}_{\mathrm{n}}+\mathrm{b}$$$$\mathrm{if}\mathrm{a}=1\iff {\mathrm{T}}_{\mathrm{n}+1}-{\mathrm{T}}_{\mathrm{n}}=\mathrm{b}\Rightarrow \underset{\mathrm{k}=1}{\sum ^{\mathrm{n}-1}}({\mathrm{T}}_{\mathrm{k}+1}-{\mathrm{T}}_{\mathrm{k}})=\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}-1}}\mathrm{b}$$$$\Rightarrow {\mathrm{T}}_{\mathrm{n}}=\mathrm{b}(\mathrm{n}-1)+{\mathrm{T}}_1$$$$\mathrm{if}\mathrm{a}\ne 1$$$$\mathrm{let}\mathrm{x}=\mathrm{ax}+\mathrm{b}\Rightarrow \mathrm{x}=\frac{\mathrm{b}}{1-\mathrm{a}}$$$$\mathrm{let}$$$${\mathrm{T}}_{\mathrm{n}}={\mathrm{W}}_{\mathrm{n}}+\frac{\mathrm{b}}{1-\mathrm{a}}$$$$\Rightarrow {\mathrm{W}}_{\mathrm{n}+1}+\frac{\mathrm{b}}{1-\mathrm{a}}=\mathrm{a}({\mathrm{W}}_{\mathrm{n}}+\frac{\mathrm{b}}{1-\mathrm{a}})+\mathrm{b}$$$$\Rightarrow {\mathrm{W}}_{\mathrm{n}+1}+\frac{\mathrm{ab}}{1-\mathrm{a}}={\mathrm{aW}}_{\mathrm{n}}+\frac{\mathrm{ab}}{1-\mathrm{a}}+\mathrm{b}={\mathrm{aW}}_{\mathrm{n}}+\frac{\mathrm{b}}{1-\mathrm{a}}$$$$\Rightarrow {\mathrm{W}}_{\mathrm{n}+1}={\mathrm{aW}}_{\mathrm{n}}\Rightarrow {\mathrm{W}}_{\mathrm{n}}={\mathrm{W}}_1{\left(\mathrm{a}\right)}^{\mathrm{n}-1}$$$${\mathrm{T}}_{\mathrm{n}}=\frac{\mathrm{b}}{1-\mathrm{a}}+{\mathrm{W}}_1\left({\mathrm{a}}^{\mathrm{n}-1}\right)$$$${\mathrm{W}}_1={\mathrm{T}}_1-\frac{\mathrm{b}}{1-\mathrm{a}}$$$$$$

Commented by TawaTawa last updated on 10/Dec/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir},\mathrm{thanks}\mathrm{for}\mathrm{your}\mathrm{time}$$

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