Arithmetic and Geometric Progressions

Size: px

Start display at page:

Download "Arithmetic and Geometric Progressions"

Ernest Freeman
6 years ago
Views:

1 GEOL Mathematical Tools in Geology Lab Assignment # 5 - Sept 30, 2008 (Due Oct 8) Name: Arithmetic and Geometric Progressions A. Arithmetic Progressions 1. We learned in class that a sequence of numbers which has an arithmetic progression can be described by the equation, b + n d. Below is an example of a sequence governed by an arithmetic progressions. Determine the value of b and the common difference (d) for each case. n AP AP If you haven t guessed, the first progression (AP1) describes temperature changes in Fahrenheit and the second progression (AP2)describes temperature changes in Celsius. If F = b + n d, write out this the equation for F with the constants you determined. Write out the equation for C as well. What is the unknown variable in the equations above? Is there a way you can combine these equations into one equation? Try to combine them to write a final equation for F that includes C. (Hint: Think about the process of substitution) 1

2 3.Look at the final equation for F that you wrote. What type of function is this? Does it look like anything familiar to you? What is the slope of this function? What is the y intercept of this function? 4. What kind of function will you obtain if you plot 2 arithmetic progressions against each other? Plot the points in the table above using AP1 (F ) for the x axis and AP2 C for the y axis. Join your points. What kind of function is this? (Try this on a computer - using Excel or in the Student computer lab (computer #5) using Octave (Matlab equivalent)). B. Pairing Arithmetic and Geometric Progressions 5. We learned in class that a sequence of numbers which has an geometric progression can be described by the equation, a r n. Below is an example of a sequence governed by an arithmetic and a geometric progression. Determine the value of b and d (common difference) for the arithmetic progression and the values of a and r (common ratio) for the geometric progression. n AP GP This particular AP expression above describes the age of a fossil sample (These particular ages were chosen to correspond to multiles of the 14 C half-life). What is the half-life of 14 C? The GP sequence describes the radioactivity of the sample in counts of beta particles on a Geiger counter per minute per gram of carbon (This assumes the 1950, pre-nuclear testing values as a starting point). 7. Write two equations, one for sample age (T), and another for counts of radioactivity (C rad ). 2

3 8. Finding the common unknowns in each equation, combine these equations to give one single equation for radioaoctivity C rad in terms of T. 9. What type of function do you obtain? Is this similar or different than the function you found in the first exercise? In general form, this should look something like y = αβ x, what type of function is this, hint: where is x located? 10. To plot this on a graph, it is most easily (and traditionally) done on a log scale. To do this, first let s work a bit with your equations. Take the log of both sides of your equation. Remember all your rules of logarithms...and do this clearly one step at a time. 11. Look at your final equation. Does this remind you of any simpler equation that you might have used before (perhaps in this lab...)? You should have 3 terms in your final equation. 3

4 12. If you could look at this equation in some way - as a linear equation, what would be the slope? What is the y intercept? 13. Let s plot your final equation on a graph. Plot the values from the table above, but plot this on a semilog scale. That is, plot the x axis (age, T) with a linear scale and plot the y axis (counts of radioactivity, C) on a logarithmic scale. You may want to do this on a computer in Excel - or in the Student computer lab using Octave - see tutorial in Octave). 14. If you join the points on your semilogy plot, what shape does the function have? What is the slope of your plot? How does this slope value relate to the common ratio, r, and the common difference, d of your original number sequences? B. Pairing Two Geometric Progressions 15. Finally we will try to pair two geometric sequences with each other. In the example below determine the common ratio, r, and a for each geometric sequence. n GP GP In the above example, GP1 is the diameter (d) of a small sphere (in cm), and GP2 is the of settling velocity (u) in cm/sec as calculated from the Stoke s Law. The Stoke s Law is one of many examples in geology a power function. The general form of this power function is y = αx β. How does this look plotted on a graph? First, look at this general power function and take the log of both sides. 17. What does your logarithmic function look like? Does it remind you of a simpler equation that you looked at in the beginning of this lab? 4

5 If you tried to look at this as a linear equation, what would be it s slope? What would be its y intercept? 18. Write out an equation for the two geometric sequences in your example. 19. Combine the equations in #18 to show a complete equation for Stokes velocity in terms of the sphere diameter (d). You should also include parameters for gravity (g), density of water (ρ w ), density of the sphere (ρ s ), and viscosity of water (µ w ). You might have to think about where these new parameters go in terms of how fast a sphere falls in a beaker of water (as you already did in your first lab in this class). 20. Plot the values of the two geometric progressions on a graph. This time use a log-log plot, that is plot both the x axis (sphere diameter,d) and y axis (Stokes velocity, u) on a log scale. You may want to do this on in Excel or on the lab computers downstairs. 21. If you joint the points on your plot, what type of function do they show? 5

6 22. How does the value for the slope of this line relate to the common ratios of each of your 2 number sequences? Some thoughts: When you look at a graph of scientific data. The first thing you should look at is the axes. Is it a linear, log, or semilog scale? Then look at the plot itself. Is it a straight line? This will give you some clue to how the variables or physical parameters relate to each other in this particular study area. 6

7 Figure 1: Lab #5: Derby Duck thermometer, floats! Figure 2: Lab #5: Fossilized wood is often used for radiometric dating 7

Section 20: Arrow Diagrams on the Integers

Section 20: Arrow Diagrams on the Integers Section 0: Arrow Diagrams on the Integers Most of the material we have discussed so far concerns the idea and representations of functions. A function is a relationship between a set of inputs (the leave