Question

What is the nth term of the geometric sequence that has a common ratio of 6 and 24 as it’s third term?

Answer 1

$\bf \begin{array}{ccll} n&term&value\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 1&a_1&a_1\\ 2&a_2&6a_1\\ 3&a_3&6(6a_1)\\ &&36a_1 \end{array}\implies \stackrel{\textit{3rd term is 24}}{36a_1=24} \\\\\\ a_1=\cfrac{24}{36}\implies a_1=\cfrac{2}{3}$

$\bf n^{th}\textit{ term of a geometric sequence} \\\\ a_n=a_1\cdot r^{n-1}\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ r=6\\ a_1=\frac{2}{3} \end{cases} \\\\\\ \boxed{a_n=\cfrac{2}{3}\cdot 6^{n-1}}$