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-

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

A geometric sequence with n terms

a_{1}, a_{2}, a_{3}, \dots, a_{n} which has a_{1} . a_{n} = 3

If the product of all n terms = a_{1} a_{2} a_{3} \dots a_{n} = 59049

Determine the value of n

Answered by linkelly0615 last updated on 29/Mar/17

\dots

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} \dots a_{n} = a_{1} (a_{1} r) (a_{1} r^{2}) \dots (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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