Name _______________________________________
Advanced Algebra
SEQUENCES AND SERIES
Period ______ Date ___________________________
Recursive Formulas for Sequences
In 1-4, write the first five terms of each sequence:
1.
 a1  3

 a n  2a n 1  1
2.
 a1  5

 a 2  1
 a  a  2a
n 2
n 1
 n
3.
 a 0  2

2
 a n  a n 1   1
4.
a1  7

a n  a n 1  n
In 5-8, (a) identify each sequence as arithmetic, geometric, or neither,
(b) find a recursive formula for the sequence.
5.
6.
13, 6,  1,  8,...
2, 2, 4, 6, 10, 16, ...
7.
36, 12, 4,
4
,...
3
8.
125, 50, 20, 8,...
You may use your calculator for the problems #9 through #12:
9.
A lake initially contains 5000 fish. Each year the population declines 20% due to fishing and other causes, and the lake is
restocked with 500 fish.
a) Find the number of fish in the lake each of the first four years. (Hint: 80% of the fish survive each year…)
b) Write a recursive formula for the fish population.
c) Use your formula to find the fish population in the lake after 10 years.
d) Is the sequence created by the fish population arithmetic, geometric, or neither? Could this problem be solved using
an explicit formula? Explain.
10. To get your swimming pool ready for the summer after the long winter, you first drop 34 ounces of chlorine into it. For
the rest of the swimming season, every week you add 16 ounces of chlorine and 40% of the chlorine evaporates.
a) Find the number of ounces of chlorine in the pool each of the first four weeks. (Hint: 60% of the chlorine stays in the
pool…)
b) Write a recursive formula for the number of ounces of chlorine in the pool.
c) Use your formula to find the amount of chlorine in the pool after 8 weeks.
d) Is the sequence created by the amount of chlorine in the pool arithmetic, geometric, or neither? Could this problem be
solved using an explicit formula? Explain.
11. Patty Patriot bought a new computer for $900 and charged it on her credit card. Her credit card company charges her
interest at a rate of 1.5% per month. She plans on making monthly payments of $50.
a) Find Patty’s balance on her credit card account each of the first four months.
b) Write a recursive formula for the balance on the credit card.
c) Use your formula to find the balance on the account one year after she purchased the computer (that is, 12 months
after the purchase.)
d) Is the sequence created by the credit card balance arithmetic, geometric, or neither? Could this problem be solved
using an explicit formula? Explain.
12. The Amazon rain forest has an estimated area of 5,500,000 square kilometers. Each year 2% of the total area is lost due
to deforestation.
a) Find the remaining area of the rain forest each of the next four years. (Hint: 98% of the rain forest is not lost…)
b) Write a recursive formula for the remaining area of the rain forest.
c) Use your formula to find the area of the rain forest 10 years from now.
d) Is the sequence created by the area of the rain forest arithmetic, geometric, or neither? Could this problem be solved
using an explicit formula? Explain.
Advanced Algebra
ANSWER KEY
SEQUENCES AND SERIES
Recursive Formulas for Sequences
1.
3.
5.
SHOW ALL YOUR WORK TO RECEIVE FULL CREDIT.
 a1  5
 a1  3

2.  a 2  1

 a n  2a n 1  1
 a  a  2a
n 2
n 1
 n
3, 5, 9, 17, 33,…
5, −1, 3, 5, 13,…
 a 0  2

2
 a n  a n 1   1
−2, 5, 26, 677, 458330,…
13, 6,  1,  8,...
Arithmetic: d = −7
4.
6.
2, 2, 4, 6, 10, 16, ...
Neither
a1  2

a2  2
a  a  a
n 1
n2
 n
8.
125, 50, 20, 8,...
a1  13

an  an 1  7
7.
36, 12, 4,
4
,...
3
Geometric: r 
a1  36


an 1
an  3
1
3
 a1  7

 a n  a n 1  n
7, 5, 2, −2, −7,…
Geometric: r 
2
5
a1  125


2
an  5 an 1
9.
A lake initially contains 5000 fish. Each year the population declines 20% due to fishing and other causes,
and the lake is restocked with 500 fish.
a) Find the number of fish in the lake each of the first four years. (Hint: 80% of the fish survive each year…)
5000, 4500, 4100, 3780
b) Write a recursive formula for the fish population.
a1  5000

an  0.8 an 1  500
c) Use your formula to find the fish population in the lake after 10 years.
About 2835 fish
d) Is the sequence created by the fish population arithmetic, geometric, or neither? Could this problem be
solved using an explicit formula? Explain.
Neither: there is not a common difference or ratio between the terms of the sequence. For this sequence we do
not know how to find an explicit formula for its nth term, so using a recursive formula is our only way to solve
this problem.
10. To get your swimming pool ready for the summer after the long winter, you first drop 34 ounces of
chlorine into it. For the rest of the swimming season, every week you add 16 ounces of chlorine and 40%
of the chlorine evaporates.
a) Find the number of ounces of chlorine in the pool each of the first four weeks.
34, 36.4, 37.84, 38.704
b) Write a recursive formula for the number of ounces of chlorine in the pool.
a1  34

an  0.6 an 1  16
c) Use your formula to find the amount of chlorine in the pool after 8 weeks.
About 39.832 ounces of chlorine
d) Is the sequence created by the amount of chlorine in the pool arithmetic, geometric, or neither? Could
this problem be solved using an explicit formula? Explain.
Neither: there is not a common difference or ratio between the terms of the sequence. For this sequence
we do not know how to find an explicit formula for its nth term, so using a recursive formula is our only
way to solve this problem.
11. Patty Patriot bought a new computer for $900 and charged it on her credit card. Her credit card company
charges her interest at a rate of 1.5% per month. She plans on making monthly payments of $50.
a) Find Patty’s balance on her credit card account each of the first four months.
900, 863.5, 826.4525, 788.8492…
b) Write a recursive formula for the balance on the credit card.
a1  900

an  1.015 an 1  50
c) Use your formula to find the balance on the account one year after she purchased the computer (that is,
12 months after the purchase.)
$466.99
d) Is the sequence created by the credit card balance arithmetic, geometric, or neither? Could this problem
be solved using an explicit formula? Explain.
Neither: there is not a common difference or ratio between the terms of the sequence. For this sequence
we do not know how to find an explicit formula for its nth term, so using a recursive formula is our only
way to solve this problem.
12. The Amazon rain forest has an estimated area of 5,500,000 square kilometers. Each year 2% of the total
area is lost due to deforestation.
a) Find the remaining area of the rain forest each of the next four years.
5,500,000; 5,390,000; 5,282,200; 5,176,556
b) Write a recursive formula for the remaining area of the rain forest.
a1  5,500, 000

an  0.98 an 1
c) Use your formula to find the area of the rain forest 10 years from now.
About 4,585,612.7 square kilometers
d) Is the sequence created by the area of the rain forest arithmetic, geometric, or neither? Could this
problem be solved using an explicit formula? Explain.
The sequence is geometric, with common ratio r = 0.98. Therefore we could have found an explicit
n 1
formula for the problem. It would be an  5,500,000   0.98 . This would allow us to find the
remaining area after 10 years much faster!