LESSON 3.13APPLICATIONS OF

ARITHMETIC SEQUENCES

Concept: Arithmetic Sequences

EQ: How do we use arithmetic sequences to solve real world problems? F.LE.2

Vocabulary: Arithmetic sequence, Common difference, Recursive formula, Explicit formula, Null/Zeroth term

THINK-WRITE-SHARE What do you know about arithmetic

sequences?

Think back to the lesson over arithmetic sequences and write down everything you remember. Be sure to include formulas.

HOW TO IDENTIFY AN ARITHMETIC SEQUENCE

In a word problem, look for a common difference being used between each term.

Example:Determine whether each situation has a

common difference 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 FINDING A TERM FOR AN ARITHMETIC SEQUENCE

1. Create a picture of the word problem. 2. Write out the sequence in order to

identify the common difference, d, 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 the explicit formula gives us a specific term.

4. Substitute d and in to the formula from step 3.

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:Picture Sequence

InterpretationFormula

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  

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

After no months have passed, the plant begins at 3 inches tall.

There is such a thing as a zeroth term or null term. It can be the initial value in certain real world examples.

Example 2:Picture Sequence

InterpretationFormula

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

25Before any weeks have passed, Kayla starts with $25.

Example 3:Picture Sequence

InterpretationFormula

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! A theater has 26 seats in row 1, 29

seats in row 2, and 32 seats in row 3 and so on. If this pattern continues, how many seats are in row 42?

You Try!Picture Sequence

InterpretationFormula

SUMMARIZER Write down 3 points that every student

should remember in order to solve arithmetic sequences in real world situations.