lesson 3.12 applications of arithmetic sequences

LESSON 3.12APPLICATIONS OF

ARITHMETIC SEQUENCES

HOW TO IDENTIFY AN ARITHMETIC SEQUENCE In a word problem, look for a constant

value being used between each term. This will indicate a common difference, d, and

an arithmetic sequence Example:

Determine whether each situation has a constant value between each term.

1. The height of a plant grows 2 inches each day.2. The cost of a video game increases by 10%

each month.3. Johnny receives 5 dollars each week for an

allowance.

STEPS TO CREATING AN EQUATION1. Create a visual representation of the

word problem using the template given.2. Identify the common difference, and

the first term, 3. Determine which formula would best fit

the situation (Recursive or Explicit)REMEMBER: Recursive formula helps us get

the next term given the previous term while explicit formula gives us a specific term.

4. Plug and in to the formula from step 2.5. Evaluate the formula for the given term.6. Interpret the result.

EXAMPLE 1 You visit the Grand Canyon and drop a

penny off the edge of a cliff.  The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence.  What is the total distance the object will fall in 6 seconds?

Example 1:Title Sequence

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EXAMPLE 1 You visit the Grand Canyon and drop a

penny off the edge of a cliff.  The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence.  What is the total distance the object will fall in 6 seconds?

1. Identify and The given sequence is 16, 48, 80, …

EXAMPLE 1 1. 2. Determine which formula would

best fit the situation. Since we want the distance after 6

seconds, we will use the explicit formula which is used to find a specific term.

Explicit Formula:

EXAMPLE 13. Plug and in to the formula from

step 2. If we plug in and from step 1, we get:

Simplify: Distribute Combine Like

Terms

EXAMPLE 13. 4. Evaluate the formula for the

given value. In the problem, we are looking for the

total distance after 6 seconds. Therefore, we will plug in 6 to the equation from step 3.

EXAMPLE 14. 5. Interpret the result

The problem referred to the total distance in feet, therefore:

After 6 seconds, the penny will have fallen a total distance of 176 feet.

EXAMPLE 2 Tom just bought a new cactus plant for

his office. The cactus is currently 3 inches tall and will grow 2 inches every month. How tall will the cactus be after 14 months?

1. Identify and

Difference between months

Height after 1 month

Example 2:Title Sequence

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EXAMPLE 2 1. 2. Determine which formula would

best fit the situation. Since we want the distance after 14

months, we will use the explicit formula which is used to find a specific term.

Explicit Formula:

EXAMPLE 23. Plug and in to the formula from

step 2. If we plug in and from step 1, we get:

Simplify: Distribute Combine Like

Terms

EXAMPLE 23. 4. Evaluate the formula for the

given value. In the problem, we are looking for the

height after 14 months. Therefore, we will plug in 14 to the equation from step 3.

EXAMPLE 24. 5. Interpret the resultThe problem referred to the height in

inches, therefore:After 14 months, the cactus will be 31

inches tall.

EXAMPLE 3 Kayla starts with $25 in her allowance

account. Each week that she does her chores, she receives $10 from her parents. Assuming she doesn’t spend any money, how much money will Kayla have saved after 1 year?

1. Identify and

Difference between weeks

Amount after 1 week

Example 3:Title Sequence

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EXAMPLE 3 1. 2. Determine which formula would

best fit the situation. Since we want the distance after 1

year (Which is ___ weeks), we will use the explicit formula which is used to find a specific term.

Explicit Formula:

EXAMPLE 33. Plug and in to the formula from

step 2. If we plug in and from step 1, we get:

Simplify: Distribute Combine Like

Terms

EXAMPLE 33. 4. Evaluate the formula for the

given value. In the problem, we are looking for the

amount after 52 weeks. Therefore, we will plug in 52 to the equation from step 3.

EXAMPLE 34.

5. Interpret the resultThe problem referred to the amount of

money, therefore:After 52 weeks, Kayla will have saved

$545.

YOU TRY! The sum of the interior angles of a

triangle is 180º, of a quadrilateral is 360º and of a pentagon is 540º.  Assuming this pattern continues, find the sum of the interior angles of a dodecagon (12 sides).

You Try!Title Sequence

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YOU TRY!

Formula:

Plug in 12

Interpret: