College Algebra, Chapter 9, 9.3, Section 9.3, Problem 46

Determine the partial sum $S_n$ of the geometric sequence that satisfies $\displaystyle a_2 = 0.12, a_5 = 0.00096$ and $n=4$.

Since this sequence is geometric, its $n$th term is given by the formula $a_n = ar^{n -1}$. Thus,

$a_2 = ar^{2-1} = ar$

$a_5 = ar^{5-1} = ar^4$

From the values we are given for these two terms, we get the following system of equations


$\left\{ \begin{equation} \begin{aligned} 0.12 =& ar \\ 0.00096 =& ar^4 \end{aligned} \end{equation} \right.$


We solve this system by dividing.


$\begin{equation} \begin{aligned} \frac{ar^4}{ar} =& \frac{0.00096}{0.12} && \\ \\ r^3 =& 0.008 && \text{Simplify} \\ \\ r =& 0.2 && \text{Take the cube root of each side} \end{aligned} \end{equation}$


Substituting for $r$ in the first equation $ar = 0.12$, gives


$\begin{equation} \begin{aligned} 0.12 =& a(0.2) \\ \\ a =& \frac{0.12}{0.2} \qquad \text{Divide by } 0.2 \\ \\ a =& \frac{3}{5} \\ \\ a =& 0.6 \end{aligned} \end{equation}$


Then using the formula for partial sum


$\begin{equation} \begin{aligned} S_n =& a \frac{1 - r^n}{1-r} \\ \\ S_4 =& 0.6 \left( \frac{1-0.2^4}{1-0.2} \right) \\ \\ S_4 =& 0.7488 \end{aligned} \end{equation}$