**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