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Real- life Geometric and Arithmetic Sequences

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Jasmine Johnson

on 20 August 2013

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Transcript of Real- life Geometric and Arithmetic Sequences

Real- life Geometric and Arithmetic Sequences
Equations
Arithmetic Sequence Situation
Your room is too cold, so you decide to adjust the thermostat. The current temperature of the room is 60˚ Fahrenheit. In an attempt to get warmer, you increase the temperature to 62˚ . When this doesn't warm the room enough for you, you decide to increase the thermostat to 64˚ . This temperature still isn't warm enough, so you continue to increase it in this manner.
While it may be difficult to see in the images provided, the temperature is being increased by 2 F each time. This created the arithmetic sequence of 60, 62, 64, 68, 70.


n = the term we are looking for
= the first term (60)
d = the common difference (2)
With this information, we can substitute our known information to create the formula
Equations (continued)
With the information previously provided, we can substitute the known information to create the formula
Visual Representation
Example
Just to demonstrate how the formulas work, let's find what the temperature would be if you adjust the thermostat starting at the original temperature, 12 times.
this will mean n= 12
Explicit and Recursive Formulas Solved
Equations (continued)
Because the sequence ends with an ellipsis, it is evident that it is infinite. Any term can be found using the explicit formula
Summation Notation
Geometric Sequence Situation
Your room is too cold so you decide to adjust the thermostat. The current temperature of the room is 60˚ Fahrenheit. In an attempt to get warmer, you increase the temperature by 10% every hour. An hour later, it's still not warm enough, so you increase it by 10% again. When this still isn't effective, you continuously increase the temperature in this manner.
Visual Representation
While it is difficult to see, each thermostat is being increased by 10%. This creates the sequence of (rounded to the nearest tenth), 60, 66, 73, 80, 88...
Equations
n = the term we are looking for
= the first term
r = common ratio
With this information, we can substitute our known information to create the formula
Equations (continued)
With the information previously provided, we can substitute our known information to create the formula
Example
Just to demonstrate how the formulas work, let's find what the temperature would be if we adjust it starting at the original temperature, 12 times.
this will mean n = 12
Equations (continued)
Summation Notation
Explicit and Recursive Formulas Solved
Because the problem says "you continuously increase..." this implies that the sequence is infinite. Any term can be found using the formula
References:
All ideas for the real-life examples were expanded from the original ideas in Laura Langhoff's video, "Geometric Sequences in the Real World" http://www.sophia.org/tutorials/geometric-sequences-in-the-real-world
Full transcript