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Question Number 11633 by Joel576 last updated on 29/Mar/17

A geometric sequence with n terms a_1 , a_2 , a_3 , ..., a_n which has a_1 . a_n = 3 If the product of all n terms = a_1 a_2 a_3 ...a_n = 59049 Determine the value of n

$$\mathrm{A}\:\mathrm{geometric}\:\mathrm{sequence}\:\mathrm{with}\:{n}\:\mathrm{terms}\: \\ $$$${a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,\:...,\:{a}_{{n}} \:\mathrm{which}\:\mathrm{has}\:{a}_{\mathrm{1}} \:.\:{a}_{{n}} \:=\:\mathrm{3} \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:{n}\:\mathrm{terms}\:=\:{a}_{\mathrm{1}} {a}_{\mathrm{2}} {a}_{\mathrm{3}} ...{a}_{{n}} =\:\mathrm{59049} \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{n} \\ $$

Answered by linkelly0615 last updated on 29/Mar/17

... Set: a_k =a_1 ∙r^(k−1) (Because the sequence is a geometric sequence.) ⇒a_1 ∙a_n =a_1 ∙(a_1 ∙r^(n−1) )=a_1 ^2 ∙r^(n−1) ∵a_k =a_1 ∙r^(k−1) ∴a_1 a_2 a_3 ...a_n =a_1 (a_1 r)(a_1 r^2 )...(a_1 r^(n−1) ) =a_1 ^n r^((((n(n−1))/2))) =(a_1 ^2 r^(n−1) )^(((n/2))) =3^(n/2) =59049 ⇒(n/2)=log _3 59049=10 (That means 3^(10) =59049) ⇒n=20

$$ \\ $$$$... \\ $$$${Set}:\:{a}_{{k}} ={a}_{\mathrm{1}} \centerdot{r}^{{k}−\mathrm{1}} \\ $$$$\left({Because}\:{the}\:{sequence}\:{is}\:{a}\:{geometric}\:{sequence}.\right) \\ $$$$\Rightarrow{a}_{\mathrm{1}} \centerdot{a}_{{n}} ={a}_{\mathrm{1}} \centerdot\left({a}_{\mathrm{1}} \centerdot{r}^{{n}−\mathrm{1}} \right)={a}_{\mathrm{1}} ^{\mathrm{2}} \centerdot{r}^{{n}−\mathrm{1}} \\ $$$$\:\because{a}_{{k}} ={a}_{\mathrm{1}} \centerdot{r}^{{k}−\mathrm{1}} \\ $$$$\therefore{a}_{\mathrm{1}} {a}_{\mathrm{2}} {a}_{\mathrm{3}} ...{a}_{{n}} ={a}_{\mathrm{1}} \left({a}_{\mathrm{1}} {r}\right)\left({a}_{\mathrm{1}} {r}^{\mathrm{2}} \right)...\left({a}_{\mathrm{1}} {r}^{{n}−\mathrm{1}} \right) \\ $$$$={a}_{\mathrm{1}} ^{{n}} {r}^{\left(\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}}\right)} =\left({a}_{\mathrm{1}} ^{\mathrm{2}} {r}^{{n}−\mathrm{1}} \right)^{\left(\frac{{n}}{\mathrm{2}}\right)} \\ $$$$=\mathrm{3}^{\frac{{n}}{\mathrm{2}}} =\mathrm{59049} \\ $$$$\Rightarrow\frac{{n}}{\mathrm{2}}=\mathrm{log}\:_{\mathrm{3}} \mathrm{59049}=\mathrm{10} \\ $$$$\left({That}\:{means}\:\mathrm{3}^{\mathrm{10}} =\mathrm{59049}\right) \\ $$$$\Rightarrow\boldsymbol{{n}}=\mathrm{20} \\ $$

Commented by Joel576 last updated on 29/Mar/17

thank you very much

$${thank}\:{you}\:{very}\:{much} \\ $$

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