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UNIT 5 TEST REVIEW: SEQUENCES AND SERIES
1. Mrs. Inscoe has square placemats made up of wooden
circles. She notices that the wooden circles are put in
diagonal lines starting in a corner. The first diagonal
“row” has 1 wooden circle. The second diagonal row has
2 wooden circles, the third row has 3, and so on. How
many wooden circles does it take to make up half of her
placemat if one half has 16 rows in it?
2. Large dogs have an average of 7 puppies in a litter. If
each of those puppies has a litter of 7 puppies, and so on
for each generation, how many puppies can trace their
lineage back to the original dog after 5 generations?
(Assume the first generation is the original litter of 7
puppies.)
ANSWER: 136 wooden circles (use arithmetic series
formula; n = 16, a1 = 1, d = 1)
ANSWER: 19,607 puppies (use geometric series
formula; n = 5, a1 = 7, r = 7)
3. Charlie collects baseball cards, and he gets new cards
each month. He starts out with no cards, and in the first
month he gets 5 cards. In the second month, he gets 10
cards; in the third month, he gets 15 cards, and so on.
How many cards does he receive in the 24th month?
4. What is the difference in a25 and S25?
ANSWER: 120 cards (use arithmetic sequence
formula because we don’t want to know how many
he gets total, just in the 24th month… n = 24, d = 5, a1
= 5)
5. How do you determine whether or not a given
sequence is arithmetic, geometric, or neither?
ANSWER: a25 is asking for the 25th term and wants
us to use a SEQUENCE formula.
S25 is asking for the SUM of the first 25 terms and
wants us to use a SERIES formula.
6. Write an example of an arithmetic sequence, a
geometric sequence, and a sequence that follows a
pattern but is neither arithmetic nor geometric.
ANSWER:
Arithmetic—add or subtract the same number every
time
Geometric—multiply by the same number every
time.
ANSWER: many correct answers. follow the rules to
the left.
example: arithmetic: 3, 5, 7, 9
geometric: 2, 6, 18, 54…
7. Write an arithmetic sequence that has four means
between 3 and 88.
8. Write a geometric sequence that has 3 means
between 256 and 81.
ANSWER: 3, 20, 37, 54, 71, 88 (use the arithmetic
sequence explicit formula to solve for d, then use it
to find the means)
ANSWER: 256, 192, 144, 108, 81 (use the geometric
sequence explicit formula to solve for r, then use it
to find the means)
9. Do the following sequences converge or diverge?
10. Does the following sequence converge or diverge?
How do you know?
an = 3(0.9)n – 1
ANSWER: Converge (geometric sequence, r < 1)
11. Does the following sequence converge or diverge?
an = (3n + 4)/n
12. Does the following sequence converge or diverge?
an = 0.8an-1 + 5
ANSWER: converge (graph it in your calculator…gets
close to horizontal)
ANSWER: Converge (generate the first 8-10 terms to
see they are getting closer to one number)
15
13. Find
2n  5
n 4
14. Find S8 of the sequence 2, 6, 18, 54, …
ANSWER: 6560 (geometric series formula)
ANSWER: 168 (use arithmetic series formula)

15. What is the 30th partial sum of the series 1 + 4 + 7 +
…
6
16. Find
n
3
7
n 2
ANSWER: 1335 (arithmetic series formula)
ANSWER: 405 (plug in 2 – 6 and add them all
together)
17. Find the sum of the infinite geometric series 4 + 7 + 
18. Find
12.25 + ….

 4(0.2)
n 1
n 1
ANSWER: can’t do it because r > 1
19. Find the common difference of the arithmetic
sequence where a1 = 6 and a31 = 276.
ANSWER: d = 9 (use arithmetic sequence explicit
formula)
21. Write the explicit and recursive formulas for the
following sequence: 240, 60, 15, 3.75…
ANSWER: 5 (infinite geometric series formula)
20. Write the explicit and recursive formula for the
following sequence: 12, 25, 38, 51…
ANSWER:
explicit: an = 12 + (n – 1)13
recursive: a1 = 12, an = an-1 + 13
22. Write the first 4 terms of the sequence an = n2 + 6
ANSWER: 7, 10, 15, 22 (plug in 1 – 4 for n)
ANSWER:
explicit: an = 240(.25)n-1
recursive: a1 = 240, an = an-1(.25)
23. Write the first 3 terms of the following sequence:
a1 = 5, an = an-1(-6)
24. Given the following recursive formula:
a1 = 7, an = an-1 + 3
Write the explicit formula that defines the same
sequence.
ANSWER: 5, -30, 180
ANSWER: an = 7 + (n – 1)(3)
25. Given the following explicit formula: an = 3(½)n – 1
Write the recursive formula that defines the same
sequence.
26. Find the sum of the first n terms of a geometric
series with a1 = 5, an= 5120, and r = 2.
ANSWER: 10235 (geometric series formula)
ANSWER: a1 = 3, an = an-1(½)
27. What is the difference between a sequence and a
series? What notation/words tell you that you are
dealing with a series?
ANSWER: sequence: list of numbers, series: sum of
the numbers in a sequence.
Sequence: a
Series: S, sigma
29. What is the common ratio of the following
sequence? 4, 10, 25, 62.5…
ANSWER: r = 2.5
28. What is the common difference of the following
sequence? 18.5, 15, 11.5, 8, 4.5…
ANSWER: -3.5 = d
30. What is the 40th term of the sequence 100, 89, 78,
67,…
ANSWER: -329 (use arithmetic sequence explicit
formula)