Find-the-nth-term-for-the-sequence-3-12-27-48-75-

Question Number 109858 by aurpeyz last updated on 26/Aug/20

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{nth}\:\mathrm{term}\:\mathrm{for}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{3}.\:\mathrm{12}.\:\mathrm{27}.\:\mathrm{48}.\:\mathrm{75}… \\ $$

Answered by Her_Majesty last updated on 26/Aug/20

$$\mathrm{3}{n}^{\mathrm{2}} \\ $$

Answered by 1549442205PVT last updated on 26/Aug/20

Since the Difference of degree 2 is constant ,the general terms has the form u_n =an^2 +bn+c.Give n the values 1,2,3...

$$\mathrm{Since}\:\mathrm{the}\:\mathrm{Difference}\:\mathrm{of}\:\mathrm{degree}\:\mathrm{2}\:\mathrm{is}\: \\ $$$$\mathrm{constant}\:,\mathrm{the}\:\mathrm{general}\:\mathrm{terms}\:\mathrm{has}\:\mathrm{the} \\ $$$$\mathrm{form}\:\:\:\:\:\mathrm{u}_{\mathrm{n}} =\mathrm{an}^{\mathrm{2}} +\mathrm{bn}+\mathrm{c}.\mathrm{Give}\:\mathrm{n}\:\mathrm{the} \\ $$$$\mathrm{values}\:\mathrm{1},\mathrm{2},\mathrm{3}… \\ $$

Commented by aurpeyz last updated on 26/Aug/20

Pls what do you mean give n the value 1 2 3? how will that give us the answer

$$\mathrm{Pls}\:\mathrm{what}\:\mathrm{do}\:\mathrm{you}\:\mathrm{mean}\:\mathrm{give}\:\mathrm{n}\:\mathrm{the}\:\mathrm{value}\:\mathrm{1}\:\mathrm{2}\:\mathrm{3}? \\ $$$$\mathrm{how}\:\mathrm{will}\:\mathrm{that}\:\mathrm{give}\:\mathrm{us}\:\mathrm{the}\:\mathrm{answer} \\ $$

Commented by 1549442205PVT last updated on 28/Aug/20

n=1⇒a.1^2 +b.1+c=3,n=2⇒a.2^2 +b.2+c=12 n=3⇒a.3^2 +b.3+c=27.We obtain a system of three eqs.of degree 1for 3 unknowns a,b ,c.Solve this system

$$\mathrm{n}=\mathrm{1}\Rightarrow\mathrm{a}.\mathrm{1}^{\mathrm{2}} +\mathrm{b}.\mathrm{1}+\mathrm{c}=\mathrm{3},\mathrm{n}=\mathrm{2}\Rightarrow\mathrm{a}.\mathrm{2}^{\mathrm{2}} +\mathrm{b}.\mathrm{2}+\mathrm{c}=\mathrm{12} \\ $$$$\mathrm{n}=\mathrm{3}\Rightarrow\mathrm{a}.\mathrm{3}^{\mathrm{2}} +\mathrm{b}.\mathrm{3}+\mathrm{c}=\mathrm{27}.\mathrm{We}\:\mathrm{obtain}\:\mathrm{a} \\ $$$$\mathrm{system}\:\mathrm{of}\:\mathrm{three}\:\mathrm{eqs}.\mathrm{of}\:\mathrm{degree}\:\mathrm{1for}\:\mathrm{3} \\ $$$$\mathrm{unknowns}\:\mathrm{a},\mathrm{b}\:,\mathrm{c}.\mathrm{Solve}\:\mathrm{this}\:\mathrm{system} \\ $$

Answered by bemath last updated on 26/Aug/20

3 ; 12; 27; 48; 75 ; ... 9; 15; 21; 27 6; 6; 6; U_n = 3 +(n−1).9 + (((n−1)(n−2))/(2!)).6 U_n =9n−6+3(n^2 −3n+2) U_n =3n^2 +9n−9n−6+6 = 3n^2

$$\mathrm{3}\:;\:\mathrm{12};\:\mathrm{27};\:\mathrm{48};\:\mathrm{75}\:;\:… \\ $$$$\:\:\:\:\mathrm{9};\:\:\:\mathrm{15};\:\:\:\mathrm{21};\:\:\:\mathrm{27} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{6};\:\:\:\:\:\mathrm{6};\:\:\:\:\:\mathrm{6};\: \\ $$$${U}_{{n}} \:=\:\mathrm{3}\:+\left({n}−\mathrm{1}\right).\mathrm{9}\:+\:\frac{\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)}{\mathrm{2}!}.\mathrm{6} \\ $$$${U}_{{n}} =\mathrm{9}{n}−\mathrm{6}+\mathrm{3}\left({n}^{\mathrm{2}} −\mathrm{3}{n}+\mathrm{2}\right) \\ $$$${U}_{{n}} =\mathrm{3}{n}^{\mathrm{2}} +\mathrm{9}{n}−\mathrm{9}{n}−\mathrm{6}+\mathrm{6}\:=\:\mathrm{3}{n}^{\mathrm{2}} \\ $$

Commented by Ari last updated on 26/Aug/20

Dear can you explain to me more clearly how you came to the general formula Un

Commented by aurpeyz last updated on 26/Aug/20

$$\mathrm{I}\:\mathrm{will}\:\mathrm{also}\:\mathrm{like}\:\mathrm{to}\:\mathrm{know}\:\mathrm{please}. \\ $$

Commented by aurpeyz last updated on 26/Aug/20

I will also like to know if i can use this approach for any kind of question?

$$\mathrm{I}\:\mathrm{will}\:\mathrm{also}\:\mathrm{like}\:\mathrm{to}\:\mathrm{know}\:\mathrm{if}\:\mathrm{i}\:\mathrm{can}\:\mathrm{use}\:\mathrm{this}\:\mathrm{approach} \\ $$$$\mathrm{for}\:\mathrm{any}\:\mathrm{kind}\:\mathrm{of}\:\mathrm{question}? \\ $$

Commented by aurpeyz last updated on 27/Aug/20

$${i}\:{tbis}\:{the}\:{formula}\:{is}\:{Newton}\:{Forward}\:{Diffrence}\:{Formula} \\ $$

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