Question

A geometric sequence is shown below : 2/3,2,6,18.. write the explicit formula

Answer 1

So we know this is a geometric sequence, meaning it has a multiplier, or a "common ratio".

now, we get the next term's value by simply multiplying the current term's value by the common ratio, so if we just divide any of the terms by the term before it, the quotient will just be the common ratio.

18/6 is just 3, so there you have it, and we know the first term is 2/3.

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