4.7 Arithmetic Sequences, Part 2: Recursive & Explicit Functions
Act 6: Explicit vs. Recursive Equations with Sequences
Noodle the cat has been put on a diet. She will only get fed 0.25 lbs. of food each day. Her owner buys a
32 lb. bag of food and begins tracking how much food is left after feeding her the first day.
days on diet
1
2
3
4
lb. food left in bag
31.75
31.5
31.25
31
a. How much food is left in the bag at the end of day 7? Show work or explain how you did this.
b. How much food is left in the bag after day 61 on the diet?
Act 7: a. Copy the table from the flipchart. Does this sequence represent an arithmetic sequence?
b. Write the explicit ‘y=’ equation for this sequence. _______________________
c.
Use your calculator to find u105. Show how you got this.
Act 8: Snowball II has begun hucking up hairballs around the house.
His owner is finding 3 new ones every month.
a. Create a table for this situation for up to 6 months.
b. Write a recursive (an) equation for this sequence. ______________________________
c. Write an explicit (y=) equation for this sequence. ______________________________
d. How many hairballs had Snowball II left around the house by the 25th month? Show work.
e. You have decided to collect the hairballs and use a needle and thread and string them to make a rope.
How long will it take you to collect 96 hairballs? Show work.
Act 9: Practice/Homework.
#1 a. Draw the next figure in the pattern of squares.
b. Write a recursive (an) equation for the sequence representing the number of squares.
c. Write an explicit equation for the sequence representing the number of squares in the pattern.
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d. Find each of the following. Show work on each using algebra.

Find the number of squares in the 47th figure.

Find the figure that will have 64 squares.
e. Check your answers from part d by entering your explicit equation into ‘y=’ on your calculator and
checking the TABLE.
#2 Use this pattern of squares for question #2.
a. The sequence for the perimeter of the squares in the pattern above, is 6, 10, 14, . . .
Prove it by counting.
b. Draw the fourth figure in the pattern of perimeters above.
c. Check to make sure that the perimeter of the fourth figure is the same as the fourth number in the
sequence: 6, 10, 14, _____, …
d. Write a recursive (an) equation for the sequence representing the perimeter of the squares.
e. What is a7? Show work.
f. Write an explicit ‘y=’ equation for this sequence. _______________________________
g. For each of the following, either show work with algebra or give a screen snapshot.

What figure will have a perimeter of 102?

What is the perimeter of the 37th figure?
(4, 6)
#3.a. What are the terms of the sequence pictured in the graph?
Make a table for this.
(3, 4)
x
y
(2,2)
b. What is u3? __________
(1, 0) (2,2)
c. If we continued the pattern, what would be the value of u5?
d. What is the common difference? How is this related to the
slope of the line through the points? (Find the slope).
Common difference: _______
Slope: ________
e. Write the recursive (an) equation for the sequence. _________________________________
f. Write the explicit (y=) equation for the sequence. _________________________________
Use your equation to complete the following ordered pairs for the sequence.
( 17, ___ )
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( 0 , ___ )
( -25 , ___ )
( ___ , 16 )
( ___ , 100)
#4 a. Write an explicit (y=) equation for an arithmetic sequence with a first term of 6.3 and a common
difference of +2.5.
b. Use the equation in #4a to figure out which term has a value of 78.8. Show work.
c. Use the equation to figure out the value of the 45 th term. Show work
#5. A sequence is defined as:
a1 = 3
an = an-1 - 4
a. Write the first 5 terms of this sequence. _____, _____, _____, _____, _____
b. Explain why this sequence is arithmetic.
c. Write an explicit (y=) equation for the sequence. ____________________________
e. Find the value of n where an = -181. Show work.
#6. After one customer buys 4 new tires, a garage recycling bin has 20 tires in it. After another
customer buys 4 new tires, the bin has 24 tires in it. (After customers get new tires put on their car, the
workers in the garage put the old tires in the recycling bin.)
a. Write a recursive (an) equation to represent the number of tires in the bin as an arithmetic sequence.
b. How many tires are in the bin after 9 customers buy all new tires? Show work.
c. Write an explicit (y=) equation that represents this situation. _________________________
d. Use your calculator to determine how many tires have been recycled after 70 customers have come in
to buy all new tires. Show work.
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