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


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

$A geometric sequence with n terms$ $a_{1}, a_{2}, a_{3}, . . ., a_{n} which has a_{1} . a_{n} = 3$ $If the product of all n terms = a_{1} a_{2} a_{3} . . . a_{n} = 59049$ $Determine the value of n$
	Answered by linkelly0615 last updated on 29/Mar/17

	$. . .$ $S e t : a_{k} = a_{1} \cdot r^{k - 1}$ $(B e c a u s e t h e s e q u e n c e i s a g e o m e t r i c s e q u e n c e .)$ $\Rightarrow a_{1} \cdot a_{n} = a_{1} \cdot (a_{1} \cdot r^{n - 1}) = a_{1}^{2} \cdot r^{n - 1}$ $∵ a_{k} = a_{1} \cdot r^{k - 1}$ $∴ a_{1} a_{2} a_{3} . . . a_{n} = a_{1} (a_{1} r) (a_{1} r^{2}) . . . (a_{1} r^{n - 1})$ $= a_{1}^{n} r^{(\frac{n (n - 1)}{2})} = {(a_{1}^{2} r^{n - 1})}^{(\frac{n}{2})}$ $= 3^{\frac{n}{2}} = 59049$ $\Rightarrow \frac{n}{2} = \log_{3} 59049 = 10$ $(T h a t m e a n s 3^{10} = 59049)$ $\Rightarrow n = 20$
	Commented by Joel576 last updated on 29/Mar/17

	$t h a n k y o u v e r y m u c h$
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