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Question Number 159221 by physicstutes last updated on 14/Nov/21

$$\mathrm{find}\mathrm{the}\mathrm{general}\mathrm{term}\mathrm{of}$$$$\mathrm{the}\mathrm{sequence}$$$$3,5,9,17,33,...$$

Commented by MJS_new last updated on 14/Nov/21

$$a_n=\frac{1}{12}{n}^4-\frac{1}{2}{n}^3+\frac{23}{12}{n}^2-\frac{3}{2}n+3$$

Commented by gsk2684 last updated on 14/Nov/21

$$plsexplainser$$

Commented by MJS_new last updated on 14/Nov/21

$$\mathrm{you}\mathrm{can}\mathrm{always}\mathrm{find}\mathrm{a}\mathrm{polynome}\mathrm{of}\mathrm{degree}$$$$k-1\mathrm{for}k\mathrm{given}\mathrm{values}.$$$${\mathrm{there}}^{\prime }\mathrm{s}\mathrm{never}\mathrm{a}\mathrm{unique}\mathrm{solution}\mathrm{for}\mathrm{a}\mathrm{question}$$$$\mathrm{like}\mathrm{this}.\mathrm{why}?\mathrm{because}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{possible}\mathrm{to}\mathrm{find}$$$$\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{general}\mathrm{term}\mathrm{for}\mathrm{any}{a}_{k+1};\mathrm{in}$$$$\mathrm{your}\mathrm{example},\mathrm{I}\mathrm{can}\mathrm{give}\mathrm{a}\mathrm{solution}\mathrm{for}$$$$3,5,9,17,33,a_6\mathrm{for}\mathrm{any}{a}_6\in \mathbb{C}$$

Commented by physicstutes last updated on 15/Nov/21

$$\mathrm{thanks}\mathrm{so}\mathrm{much}\mathrm{sirs}$$

Answered by qaz last updated on 14/Nov/21

$${\mathrm{a}}_{\mathrm{n}+1}-{\mathrm{a}}_{\mathrm{n}}=2^{\mathrm{n}}$$$$\Rightarrow \underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}({\mathrm{a}}_{\mathrm{k}+1}-{\mathrm{a}}_{\mathrm{k}})=\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}{2}^{\mathrm{k}}$$$$\Rightarrow {\mathrm{a}}_{\mathrm{n}+1}=\frac{2(1-2^{\mathrm{n}})}{1-2}+{\mathrm{a}}_1=2^{\mathrm{n}+1}-2+3=2^{\mathrm{n}+1}+1$$$$\Rightarrow {\mathrm{a}}_{\mathrm{n}}=2^{\mathrm{n}}+1$$

Answered by floor(10²Eta[1]) last updated on 15/Nov/21

$${\mathrm{a}}_1={\mathrm{a}}_1$$$${\mathrm{a}}_2={\mathrm{a}}_1+2$$$${\mathrm{a}}_3={\mathrm{a}}_2+2^2$$$${\mathrm{a}}_4={\mathrm{a}}_3+2^3$$$$...$$$${\mathrm{a}}_{\mathrm{n}}={\mathrm{a}}_{\mathrm{n}-1}+2^{\mathrm{n}-1}$$$${\mathrm{S}}_{\mathrm{n}}={\mathrm{a}}_1+({\mathrm{S}}_{\mathrm{n}}-{\mathrm{a}}_{\mathrm{n}})+(2+2^2+...+2^{\mathrm{n}-1})$$$${\mathrm{a}}_{\mathrm{n}}={\mathrm{a}}_1+2^{\mathrm{n}}-2$$$${\mathrm{a}}_{\mathrm{n}}=2^{\mathrm{n}}+1$$

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