10/6/2015
Recursive Formulas
For Arithmetic
Sequences
10/6/2015
• If you buy a new car, you might be advised
to have an oil change after driving 1000
miles and every 3000 miles thereafter.
Then the following sequence gives the
mileage when oil changes are required:
1000 4000 7000 10000 13000 16000
10/6/2015
Arithmetic Sequences
Arithmetic Sequence:
– Sequence with a constant difference between
terms. Here it was d = 3000
Recursive Formula:
– Formula where each term is based on the
term before it
Recursive Formula for an Arithmetic Seq:
a1
a  a  d , n  2
 n n1
10/6/2015
• If you buy a new car, you might be advised
to have an oil change after driving 1000
miles and every 3000 miles thereafter.
Then the following sequence gives the
mileage when oil changes are required:
1000 4000 7000 10000 13000 16000
a1  1000
a  a  3000; n  2
 n n1
10/6/2015
Example
•
Consider the sequence generated by
a1  2000
a  a  40, n  2
 n n1
A) Describe the sequence in words
B) Write the 1st five terms of the sequence
10/6/2015
Example
• Briana borrowed $870 from her parents for
airfare to Europe. She will pay them back
at the rate of $60.00 per month. Let an be
the amount she still owes after n months.
Find a recursive formula for this sequence.
a1  870
a  a  60, n  2
 n n1
10/6/2015
Graph of an Arith. Seq.
• Discrete Domain
• Constant Increase or
Decrease
• Collinear Points
a1  1
a  a  2, n  2
 n n1
10/6/2015
Explicit Formulas
For Arithmetic
Sequences
10/6/2015
Arithmetic Sequences
Explicit Formula
– Formula where any term can be found by
substituting the number of that term.
– We can develop an explicit formula for an
Arithmetic Sequence from the recursive
formula
10/6/2015
Explicit Formula
a1  1000
a  a  3000; n  2
 n n1
n
1
2
3
4
an
1000
1000+3000=4000
4000+3000=7000
7000+3000=10000
an  a1  (n  1)d
# of d
0
1
2
3
10/6/2015
• So, for our oil change example, the explicit
formula looks like:
an  1000  (n  1)3000
10/6/2015
Examples
1. Find the 40th term of the arithmetic
sequence 100,97,94,91,…..
2. In a concert hall the 1st row has 20 seats
in it, and each subsequent row has 2
more seats than the row in front of it. If
the last row has 64 seats, how many
rows are in the concert hall?