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We may use a recursively defined sequence to approximate the current amount of a radioactive element. For example, radioactive radium changes into lead over time. The rate of decay is proportional to the amount of radium present. Experimental data suggests that a gram of radium decays into lead at a rate of $\frac{1}{2337}$gram per year. Let ak be the amount of radium at the end of year k. Since the decay rate is constant, if we use a linear model to approximate the amount that remains after one year has passes, we have

role="math" localid="1649307898272" $a1=a0−\frac{1}{2337}=\frac{2336}{2337}a0.$

More generally, we obtain the recursion formula

$ak+1=\frac{2336}{2337}ak.$

Use this formula to estimate how much radium remains after 100 years if we start off with a 0 = 10 grams of radium.

Step by step solution

01

Step 1. Given information

$a1=a0−\frac{1}{2337}=\frac{2336}{2337}a0$

02

Step 2. Calculate ak+1 for k=0

$\begin{array}{r}{a}_1=\frac{2336}{2337}{a}_0\\ =\frac{2336}{2337}\left(10\right)\\ =10\left(\frac{2336}{2337}\right)\end{array}$

03

Step 3. Calculateak+1 for k=1

$\begin{array}{r}{a}_1=\frac{2336}{2337}\left(\frac{2336×10}{2337}\right)\\ =10{\left(\frac{2336}{2337}\right)}^2\end{array}$

04

Step 3. Calculate for 100 years

$\begin{array}{r}{a}_{99+1}=\frac{2336}{2337}{a}_{99}\\ a_{100}=\frac{2336}{2337}\left(10{\left(\frac{2336}{2337}\right)}^{100}\right)\\ =10{\left(\frac{2336}{2337}\right)}^{101}\end{array}$

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