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}
$
Wednesday, January 20, 2016
College Algebra, Chapter 9, 9.3, Section 9.3, Problem 46
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