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Question Number 206532 by necx122 last updated on 17/Apr/24

$$Findthenthtermofthesequence$$$$0,3,8,15,24,......$$

Answered by Berbere last updated on 17/Apr/24

$$Impossibl$$$$thereare[infinityway$$$$n^2-1;n\in Z_{+}oneofthem$$

Answered by BaliramKumar last updated on 18/Apr/24

$$0\underset{+3}{\underbrace{}}3\underset{+5}{\underbrace{}}8\underset{+7}{\underbrace{}}15\underset{+9}{\underbrace{}}24.........$$$$a=3d=5-3=2$$$${\mathrm{S}}_{\mathrm{n}-1}=\frac{\mathrm{n}-1}{2}[2\times 3+(\mathrm{n}-1-1)\times 2]$$$${\mathrm{S}}_{\mathrm{n}-1}=(\mathrm{n}-1)[3+(\mathrm{n}-1-1)]$$$${\mathrm{S}}_{\mathrm{n}-1}=(\mathrm{n}-1)(\mathrm{n}+1)={\mathrm{n}}^2-1$$$${\mathrm{U}}_1+{\mathrm{S}}_{\mathrm{n}-1}={\mathrm{U}}_{\mathrm{n}}=0+{\mathrm{n}}^2-1$$$${\mathrm{U}}_{\mathrm{n}}={\mathrm{n}}^2-1$$

Commented by necx122 last updated on 18/Apr/24

$$wow!Thidissoclear.Thankyou.$$

Answered by Frix last updated on 17/Apr/24

$$\mathrm{I}\mathrm{get}\mathrm{the}\mathrm{very}\mathrm{nice}$$$$u_n=(n-1)!-\frac{(n-5)}{24}(9n^3-37n^2+70n-48)$$$$\mathrm{or}\mathrm{if}\mathrm{you}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{like}n!$$$$u_n=\frac{1}{2}(2^n-\frac{n^4}{12}+\frac{n^3}{2}+\frac{n^2}{12}+\frac{3n}{2}-4)$$

Commented by BaliramKumar last updated on 18/Apr/24

$$\mathrm{how}\mathrm{to}\mathrm{find}?$$

Commented by Frix last updated on 18/Apr/24

$$\mathrm{Just}\mathrm{be}\mathrm{creative}.$$

Commented by BaliramKumar last updated on 19/Apr/24

$$\mathrm{not}\mathrm{true}\mathrm{for}\mathrm{n}\ge 6$$

Commented by mr W last updated on 19/Apr/24

$$onlythefirst5numbersaregiven.$$$$thenumbersuponthe{6}^{th}canbe$$$$anynumbers.thatmeanssucha$$$$questionmayhaveinfinitelymany$$$$solutions.thereisnoreasonfor$$$$youtosaythatthe{6}^{th}mustbe35.$$

Commented by Frix last updated on 19/Apr/24

$$\mathrm{Usually}\mathrm{we}\mathrm{search}\mathrm{for}\mathrm{a}\mathrm{polynome}:$$$$n\mathrm{numbers}\mathrm{given}\Rightarrow \mathrm{polynome}\mathrm{of}\mathrm{degree}n-1$$$$\mathrm{In}\mathrm{the}\mathrm{following}\mathrm{cases}\mathrm{this}\mathrm{gives}\mathrm{the}\mathrm{red}$$$$\mathrm{next}\mathrm{numbers}:$$$$0,1,2$$$$0,1,1,0$$$$0,1,1,2,6$$$$0,1,1,2,3,0$$$$0,1,1,2,3,5,16$$$$0,1,1,2,3,5,8,0$$$$...$$$$\mathrm{Think}\mathrm{about}\mathrm{it}!$$

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