Chapter 7.8, Problem 34PPE

To determine

To find:The explicit and recursive formulas for the geometric sequence formed by the amount of Fl-18 remaining.

Answer to Problem 34PPE

The explicit formula is $a_n=260\cdot {0.5}^{n-1}$

The recursive formula is $a_1=260;a_n=a_{n-1}\cdot 0.5$

Explanation of Solution

Given information:When a radioactive substance decays, measurements of the amount remaining over constant intervals of time form a geometric sequence. The table shows the amount of Fl-18, remaining after different constant intervals.

  

Calculation:

The geometric sequence is $260,130,65,32.5,\mathrm{...}$

The first term $a_1$ is 260 and the common ratio r is $\frac{130}{260}=0.5$

The explicit formula is given by,

  $\begin{array}{l}{a}_n=a_1\cdot r^{n-1}\\ \Rightarrow a_n=260\cdot r^{n-1}[\text{Substituting the first term }{a}_1\text{=260]}\\ \Rightarrow a_n=260\cdot {0.5}^{n-1}[\text{Substituting the common ratio }r=0.5]\end{array}$

Thus, the explicit formula is $a_n=260\cdot {0.5}^{n-1}$ .

The recursive formula is given by,

  $\begin{array}{l}{a}_1=a;a_n=a_{n-1}\cdot r\\ \Rightarrow a_1=260;a_n=a_{n-1}\cdot r[\text{Substituting the first term }{a}_1\text{=260]}\\ \Rightarrow a_1=260;a_n=a_{n-1}\cdot 0.5[\text{Substituting the common ratio }r=0.5]\end{array}$

Thus, the recursive formula is $a_1=260;a_n=a_{n-1}\cdot 0.5$