Applications of Sequence and Series
Question
1. An auditorium has 20 seats on the first
row, 24 seats on the second row, 28
seats on the third row, and so on and
has 30 rows of seats. How many seats
are in the theatre?
2. Logs are stacked in a pile with 24 logs
on the bottom row and 15 on the top
row. There are 10 rows in all with each
row having one more log than the one
above it. How many logs are in the
stack?
3. Each hour, a grandfather clock chimes
the number of times that corresponds
to the time of day. For example, at
3:00, it will chime 3 times. How many
times does the clock chime in a day?
4. A ball is dropped from a height of 16
feet. Each time it drops, it rebounds
80% of the height from which it is
falling. Find the height of the 7th
bounce.
5. A ball is dropped from a height of 16
feet. Each time it drops, it rebounds
80% of the height from which it is
falling. Find the total distance traveled
in 15 bounces.
6. A bookshelf has 7 shelves of different
widths. Each shelf is narrower than the
shelf below it. The bottom three
shelves are 36 in., 31 in., and 26 in.
wide. What is the width of the top
shelf?
7. A bookshelf has 7 shelves of different
widths. Each shelf is narrower than the
shelf below it. The bottom three
shelves are 36 in., 31 in., and 26 in.
wide. What is the total shelf space of all
seven shelves?
Term
or
Sum?
A
or
G?
Formula and Filled In
Answer
8. An embroidery pattern calls for five
stitches in the first row and for three
more stitches in each successive row.
How many stitches are in the 25th row,
which is the last row?
9. An embroidery pattern calls for five
stitches in the first row and for three
more stitches in each successive row.
Find the total number of stitches in the
entire 25 row pattern.
10. The deer population in an area is
increasing. This year, the population
was 1.025 times last year’s population
of 2537. Assume that the pattern
continues geometrically. What is the
expected deer population in the 5th
year?
11. To train new employees, an employer
offers a bonus after 30 work days as
follows: an employee must turn in one
report on the first day; the number of
reports for each subsequent day must
increase by two. What is the minimum
number of reports an employee will
have to turn in over the 30 days to earn
the bonus?
12. Each year, a volunteer organization expects to add 5 more people to the number of shut-ins for whom the group
provides home maintenance services. This year, the organization provides the service for 32 people.
a. Write a recursive formula for the number of people the organization expects to serve each year.
b. Write the first five terms of the sequence.
c. Write an explicit formula for the number of people the organization expects to serve each year.
d. How many people would the organization expect to serve in the 20th year?
Applications of Sequence and Series KEY
Question
1. An auditorium has 20 seats on the first
row, 24 seats on the second row, 28
seats on the third row, and so on and
has 30 rows of seats. How many seats
are in the theatre?
20, 24, 28, 32, 36, …
2. Logs are stacked in a pile with 24 logs
on the bottom row and 15 on the top
row. There are 10 rows in all with each
row having one more log than the one
above it. How many logs are in the
stack?
24, 23, 22, 21, …, 15
3. Each hour, a grandfather clock chimes
the number of times that corresponds
to the time of day. For example, at
3:00, it will chime 3 times. How many
times does the clock chime in a day?
1, 2, 3, … , 12, 1, 2, 3, … , 12
4. A ball is dropped from a height of 16
feet. Each time it drops, it rebounds
80% of the height from which it is
falling. Find the height of the 7th
bounce. (start with first bounce)
12.8, 10.24, 8.192, …
5. A ball is dropped from a height of 16
feet. Each time it drops, it rebounds
80% of the height from which it is
falling. Find the total distance traveled
in 15 bounces.
Think one set of downs and one set of ups.
downs: 16, 12.8, 10.24, …
ups: 12.8, 10.24, 8.192, …
Notice they are the same except for the 16.
6. A bookshelf has 7 shelves of different
widths. Each shelf is narrower than the
shelf below it. The bottom three
shelves are 36 in., 31 in., and 26 in.
wide. What is the width of the top
shelf?
36, 31, 26, 21, …
7. A bookshelf has 7 shelves of different
widths. Each shelf is narrower than the
shelf below it. The bottom three
shelves are 36 in., 31 in., and 26 in.
wide. What is the total shelf space of all
seven shelves?
36, 31, 26, 21, …
Term
or
Sum?
Sum
Sum
Sum
A
or
G?
A
A
A
Formula and Filled In
𝑎1 = 20; 𝑑 = 4; 𝑛 = 30
(𝑎1 + 𝑎𝑛 )𝑛
𝑆𝑛 =
2
(20 + 20 + 29(4))30
𝑆30 =
2
𝑎1 = 24; 𝑑 = 1; 𝑛 = 10; 𝑎10 = 15
(𝑎1 + 𝑎𝑛 )𝑛
𝑆𝑛 =
2
(24 + 15)10
𝑆10 =
2
𝑎1 = 1; 𝑑 = 1; 𝑛 = 12; 𝑎12 = 12
(𝑎1 + 𝑎𝑛 )𝑛
𝑆𝑛 =
2
(1 + 12)12
𝑆12 = 2 [
]
2
Answer
2340 seats
195 logs
156 chimes
𝑎1 = 12.8; 𝑟 = 0.8; 𝑛 = 7
𝑎𝑛 = 𝑎1 (𝑟)𝑛−1
Term
G
𝑎7 = 12.8(0.8)6
3.355 ft
𝑎1 = 12.8; 𝑟 = 0.8; 𝑛 = 15
𝑆𝑛 =
Sum
G
𝑆15 =
𝑎1 (1 − 𝑟 𝑛 )
1−𝑟
12.8(1 − 0.815 )
1 − 0.8
139.497 ft
Need 16 + 2𝑆𝑛
𝑎1 = 36; 𝑑 = −5; 𝑛 = 7
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
Term
A
𝑎7 = 36 + (6)(−5)
6 in
𝑎1 = 36; 𝑑 = −5; 𝑛 = 7; 𝑎𝑛 = 6
(𝑎1 + 𝑎𝑛 )𝑛
2
(36 + 6)7
𝑆7 =
2
𝑆𝑛 =
Sum
A
147 in
8. An embroidery pattern calls for five
stitches in the first row and for three
more stitches in each successive row.
How many stitches are in the 25th row,
which is the last row?
5, 8, 11, 14, …
9. An embroidery pattern calls for five
stitches in the first row and for three
more stitches in each successive row.
Find the total number of stitches in the
entire 25 row pattern.
5, 8, 11, 14, …
10. The deer population in an area is
increasing. This year, the population
was 1.025 times last year’s population
of 2537. Assume that the pattern
continues geometrically. What is the
expected deer population in the 5th
year?
2537, 2600.425, 2665.436, …
11. To train new employees, an employer
offers a bonus after 30 work days as
follows: an employee must turn in one
report on the first day; the number of
reports for each subsequent day must
increase by two. What is the minimum
number of reports an employee will
have to turn in over the 30 days to earn
the bonus?
1, 3, 5, 7, …
𝑎1 = 5; 𝑑 = 3; 𝑛 = 25
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
Term
A
𝑎25 = 5 + (24)(3)
77 stitches
𝑎1 = 5; 𝑑 = 3; 𝑛 = 25; 𝑎𝑛 = 77
Sum
(𝑎1 + 𝑎𝑛 )𝑛
2
(5 + 77)25
=
2
𝑆𝑛 =
A
𝑆25
1025
stitches
𝑎1 = 2537; 𝑟 = 1.025; 𝑛 = 5
𝑎𝑛 = 𝑎1 (𝑟)𝑛−1
Term
𝑎5 = 2537(1.025)4
G
Sum
A
𝑎1 = 1; 𝑑 = 2; 𝑛 = 30
(𝑎1 + 𝑎𝑛 )𝑛
𝑆𝑛 =
2
(1 + 1 + 29(2))30
𝑆30 =
2
2800 deer
900 reports
12. Each year, a volunteer organization expects to add 5 more people to the number of shut-ins for whom the group
provides home maintenance services. This year, the organization provides the service for 32 people.
a. Write a recursive formula for the number of people the organization expects to serve each year.
𝑎1 = 32; 𝑎𝑛 = 𝑎𝑛−1 + 5
b. Write the first five terms of the sequence.
32, 37, 42, 47, 52, …
c. Write an explicit formula for the number of people the organization expects to serve each year.
𝑎𝑛 = 5𝑛 + 27
d. How many people would the organization expect to serve in the 20th year?
𝑎1 = 32; 𝑑 = 5; 𝑛 = 20
𝑆𝑛 =
𝑆20 =
(𝑎1 + 𝑎𝑛 )𝑛
2
(32 + 32 + 19(5))20
= 1590 people
2