Arithmetic Sequences (Recursive Formula)
Objectives:
To write explicit (and recursive) formulas for arithmetic sequences.
To use recursive and explicit formulas to find terms in a sequence.
To determine if a given number is part of a sequence.
To graph arithmetic sequences.
Review of Explicit Formula:
Writing a Recursive Formula for an Arithmetic Sequence
1. Determine that the sequence is arithmetic.
2. Identify the common difference.
3. Create a recursive formula using the first term in the sequence and the common difference.
Example #1:
7, 10, 13, 16, …
a1 = 7
d=3
Explicit formula:
Recursive Formula:
𝑎𝑛 = 7 + 3(𝑛 − 1)
𝑎𝑛 = 7 + 3𝑛 − 3
𝑎𝑛 = 3𝑛 + 4
𝑎1 = 7
𝑎𝑛 = 𝑎𝑛−1 + 3
Example #2:
386, 365, 344, 323, …
Explicit formula:
a1 = 386
d = -21
Recursive formula:
𝑎𝑛 = 386 − 21(𝑛 − 1)
𝑎𝑛 = 386 − 21𝑛 + 21
𝑎𝑛 = −21𝑛 + 407
𝑎𝑛 = 386
𝑎𝑛 = 𝑎𝑛−1 − 21
Practice Problems:
Write both an explicit and a recursive formula for the following sequences.
1.
0, -3, -6, -9, …
2.
3, 8, 13, 18,
3.
0.9, 0.5, 0.1, -0.3, …
4.
3.2, 3.5, 3.8, 4.1, …
Arithmetic Sequences (Recursive Formula)
Application Problem:
Suppose you participate in a bike-a-thon for charity. The charity starts with $1100 in
donations. Each participant must raise at least $35 in pledges. What is the minimum
amount of money raised if there are 75 participants?
In other words, “What is the 75th term in the sequence?”
*Be careful identifying the first term in the sequence!*
The term number will represent the number of participants. How much money does the
charity have with 0 participants? With 1 participant?
1100,
𝑎0
1135,
𝑎1
1170,
𝑎2
…
For a problem
like this, it is a
𝑎𝑛 = 1135 + 35(𝑛 − 1)
𝑎𝑛 = 1135 + 35𝑛 − 35
𝑎𝑛 = 1100 + 35𝑛
Explicit Formula:
better idea to
use the explicit
𝑎1 = 1135
𝑎𝑛 = 𝑎𝑛−1 + 35
Recursive Formula:
formula.
Determining if a Number is Part of a Sequence
Example #1:
Example#2:
Is the number 27 a term in the sequence represented by the
explicit formula: 𝑎𝑛 = 4𝑛 − 1?
Is the number 97 a term in the sequence represented by the
explicit formula: 𝑎𝑛 = 4𝑛 − 1?
27 = 4𝑛 − 1
+1
+1
28 = 4𝑛
7=𝑛
27 is the 7th term in the sequence!
97 = 4𝑛 − 1
+1
+1
98 = 4𝑛
24.5 = 𝑛
97 is NOT in the sequence because it falls between the 24th
and 25th terms!
Application (Going Backwards):
Ellen borrowed $370 from her parents. She will pay them back at the rate of $60 per month. How long will it take for her to
pay her parents back?
Generate the sequence:
370,
𝑎0
310,
𝑎1
250,
𝑎2
…
𝑎1 = 310
Write an equation:
Explicit:
𝑎𝑛 = 310 − 60(𝑛 − 1)
𝑎𝑛 = −60𝑛 + 370
Recursive:
𝑎1 = 310
𝑎𝑛 = 𝑎𝑛−1 − 60
Solve and answer the question:
0 = −60𝑛 + 370
−370 = −60𝑛
−60
− 60
1
6 =𝑛
370, 310, 250, 190, 130, 70, 10, -50
after 7 months
6
Ellen will be done repaying her parents in the
𝑑 = −60
7 th
month.
For a problem like this, we could use EITHER
an explicit or recursive formula.