The document discusses arithmetic sequences and their applications in real-world problems, including defining key vocabulary and formulas. It provides several examples and step-by-step instructions on identifying common differences and creating explicit or recursive formulas to solve specific problems. The importance of understanding initial values and evaluating results based on these sequences is emphasized.

1. 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

2.  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.

3.  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.

4. 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, 𝒂𝟏.
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 𝑎1 in to the formula from
step 3.
5. Evaluate the formula for the given term.
6. Interpret the result.

5.  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?

6. Example 1:
Picture Sequence
Interpretation
Formula

7.  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 𝑑 and 𝑎1
The given sequence is 16, 48, 80, …
d = 48 − 16 = 32

9. 3. Plug 𝑑 and 𝑎1 in to the formula from
step 2.
 If we plug in 𝑑 and 𝑎1 from step 1, we get:
𝐴𝑛 = 32 𝑛 − 1 + 16
Simplify: 𝐴 𝑛= 32𝑛 − 32 + 16 Distribute
𝐴𝑛 = 32𝑛 − 16 Combine Like
Terms

10. 3. 𝐴𝑛 = 32𝑛 − 16
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.
𝐴𝑛 = 32𝑛 − 16
𝐴6 = 32(6) − 16
𝐴6 = 192 − 16
𝐴6 = 176

11. 4. 𝐴6 = 176
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.

12.  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 𝑑 and 𝑎1
𝑑 = 2 𝑎1 = 5
Difference
between
months
Height after 1
month
𝑎0 = 3
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.

13. Example 2:
Picture Sequence
Interpretation
Formula

14. 1. 𝑑 = 2 𝑎𝑛𝑑 𝑎1 = 5
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:
𝐴𝑛 = 𝑑 𝑛 − 1 + 𝐴1

15. 3. Plug 𝑑 and 𝑎1 in to the formula from
step 2.
 If we plug in 𝑑 and 𝑎1 from step 1, we get:
𝐴𝑛 = 2 𝑛 − 1 + 5
Simplify: 𝐴 𝑛= 2𝑛 − 2 + 5 Distribute
𝐴𝑛 = 2𝑛 + 3 Combine Like
Terms

16. 3. 𝐴𝑛 = 2𝑛 + 3
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.
𝐴𝑛 = 2𝑛 + 3
𝐴14 = 2 14 + 3
𝐴14 = 28 + 3
𝐴14 = 31

17. 4. 𝐴14 = 31
5. Interpret the result
The problem referred to the height in
inches, therefore:
After 14 months, the cactus will be 31
inches tall.

18.  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 𝑑 and 𝑎1
𝑑 = 10 𝑎1 = 35
Difference between
weeks
Amount after 1
week
𝑎0 =25
Before any
weeks have
passed, Kayla
starts with $25.

19. Example 3:
Picture Sequence
Interpretation
Formula

20. 1. 𝑑 = 10 𝑎𝑛𝑑 𝑎1 = 35
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:
𝐴𝑛 = 𝑑 𝑛 − 1 + 𝐴1

21. 3. Plug 𝑑 and 𝑎1 in to the formula from
step 2.
 If we plug in 𝑑 and 𝑎1 from step 1, we get:
𝐴𝑛 = 10 𝑛 − 1 + 35
Simplify: 𝐴 𝑛= 10𝑛 − 10 + 35 Distribute
𝐴𝑛 = 10𝑛 + 25 Combine Like
Terms

22. 3. 𝐴𝑛 = 10𝑛 + 25
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.
𝐴𝑛 = 10𝑛 + 25
𝐴52 = 10 52 + 25
𝐴52 = 520 + 25
𝐴52 = 545

23. 4. 𝐴52 = 545
5. Interpret the result
The problem referred to the amount of
money, therefore:
After 52 weeks, Kayla will have saved $545.

24.  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?

25. You Try!
Picture Sequence
Interpretation
Formula

26.  Write down 3 points that every student
should remember in order to solve arithmetic
sequences in real world situations.