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Writing and Using Explicit Rules for Geometric Sequences

Writing and Using Explicit Rules for Geometric Sequences 1.5 - Solution

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The explicit rule of a geometric sequence is given below. an=a1rn1\begin{aligned} a_n={\color{#0000FF}{a_1}} \cdot {\color{#FF0000}{r}}^{n-1} \end{aligned} Here, a1{\color{#0000FF}{a_1}} is the initial value, and r{\color{#FF0000}{r}} is the common ratio. The initial value is 18,{\color{#0000FF}{18}}, and the common ratio is 12,{\color{#FF0000}{\frac{1}{2}}}, so we can substitute. an=18(12)n1\begin{aligned} a_n={\color{#0000FF}{18}} \cdot \left( {\color{#FF0000}{\dfrac{1}{2}}}\right)^{n-1} \end{aligned} We want to write a function representing the sequence. To do so, we will substitute f(x){\color{#009600}{f(x)}} for an{\color{#009600}{a_n}} and x\textcolor{darkviolet}{x} for n.\textcolor{darkviolet}{n}. an=18(12)n1f(x)=18(12)x1\begin{aligned} {\color{#009600}{a_n}}={\color{#0000FF}{18}} \cdot \left( {\color{#FF0000}{\dfrac{1}{2}}}\right)^{\textcolor{darkviolet}{n}-1} \quad \Leftrightarrow \quad {\color{#009600}{f(x)}}={\color{#0000FF}{18}} \cdot \left( {\color{#FF0000}{\dfrac{1}{2}}}\right)^{\textcolor{darkviolet}{x}-1} \end{aligned} The difference between ana_n and f(x)f(x) is that they have different domains. The domain of the function is all real numbers whereas the domain of the sequence is the positive integers greater than or equal to 1.1. To graph the function, let's make a table of values.

xx 18(12)x118 \cdot \left(\frac{1}{2} \right)^{x-1} f(x)=18(12)x1f(x)=18 \cdot \left(\frac{1}{2} \right)^{x-1}
1\textcolor{darkviolet}{1} 18(12)1118 \cdot \left(\frac{1}{2} \right)^{\textcolor{darkviolet}{1}-1} 18{\color{#009600}{18}}
2\textcolor{darkviolet}{2} 18(12)2118 \cdot \left(\frac{1}{2} \right)^{\textcolor{darkviolet}{2}-1} 9{\color{#009600}{9}}
3\textcolor{darkviolet}{3} 18(12)3118 \cdot \left(\frac{1}{2} \right)^{\textcolor{darkviolet}{3}-1} 4.5{\color{#009600}{4.5}}
4\textcolor{darkviolet}{4} 18(12)4118 \cdot \left(\frac{1}{2} \right)^{\textcolor{darkviolet}{4}-1} 2.25{\color{#009600}{2.25}}

The points (1,18),(\textcolor{darkviolet}{1},{\color{#009600}{18}}), (2,9),(\textcolor{darkviolet}{2},{\color{#009600}{9}}), (3,4.5),(\textcolor{darkviolet}{3},{\color{#009600}{4.5}}), and (4,2.25)(\textcolor{darkviolet}{4},{\color{#009600}{2.25}}) all lie on the function f(x).f(x). Let's plot and connect them with a smooth curve.