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How many terms are there in a geometric series if the first term is 2, the common ratio is 3, and the sum of the series is 728?

How many terms are there in a geometric series if the first term is 2 the common ratio is 3 and the sum of the series is 728 class=

Respuesta :

[tex]\bf \qquad \qquad \textit{sum of a finite geometric sequence}\\\\ S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ a_1=2\\ r=3 \end{cases} \\\\\\ 728=2\left( \cfrac{1-3^n}{1-3} \right)\implies 728=\cfrac{\underline{2}(1-3^n)}{-\underline{2}}\implies 728=-(1-3^n) \\\\\\ 728=3^n-1\implies 729=3^n\qquad \boxed{729=3^6}\qquad 3^6=3^n\implies 6=n[/tex]

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