Question

Determine o número de termos da P.G. (1, 3, 9,.., 729.)

(a) 10
(b) 9
(c) 8
(d) 7

Answer 1

$\Large\boxed{\begin{array}{l}\underline{\sf Modo\,\mathbb{T}\mathbb{I}\mathbb{T}\mathbb{A}\mathbb{N}}\\\rm(1,3,9,\dotsc729)\\\rm q=\dfrac{3}{1}=3\\\\\rm a_n=a_1\cdot q^{n-1}\longrightarrow n=\ell og_q\bigg(\dfrac{a_n}{a_1}\bigg)+1\\\\\rm n=\ell og_3\bigg(\dfrac{729}{1}\bigg)+1\\\\\rm n=\ell og_{3^1}3^{6}+1\\\\\rm n=\dfrac{6}{1}+1\\\\\rm n=6+1\\\rm n=7\\\huge\boxed{\boxed{\boxed{\boxed{\rm\dagger\red{\maltese}~\blue{alternativa~d}}}}}\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\sf Modo\,easy:}\\\begin{cases}\sf a_1=1\\\sf q=\dfrac{3}{1}=3\\\sf a_n=729\\\sf n=?\end{cases}\\\underline{\sf soluc_{\!\!,}\tilde ao\!:}\\\rm a_n=a_1\cdot q^{n-1}\\\rm729=1\cdot 3^{n-1}\\\rm 3^{n-1}=729\\\rm 3^{n-1}=3^6\\\rm n-1=6\\\rm n=6+1\\\rm n=7\\\huge\boxed{\boxed{\boxed{\boxed{\rm\dagger\red{\maltese}~\blue{alternativa~d}}}}} \end{array}}$