Name: _____________________________________
Date: ____________________
Arithmetic Sequences
Warm Up: Consider the following sequence whose nth term is given by 𝑓(𝑛): 5, 8, 11, 14, …
Write a rule for this sequence using function notation and subscript notation.
Find the equation of the line passing through the points (1, 5) and (2, 8).
Arithmetic Sequences
An arithmetic sequence is a sequence such that there exist a common difference between each of
the terms. We use the variable d to denote the common difference. In simpler terms, a sequence
is arithmetic if we are adding/subtracting the same amount between every term.
Explicitly defined, arithmetic sequences are simply linear functions.
1. For each of the following sequences, determine if it is arithmetic. Fill in the missing term if it is
arithmetic. For the arithmetic sequences, write a rule in subscript notation and function
notation.
a.) 5, 9, 13, ___________, 21, 25
b.) 5, 10, 20, __________, 80, 160
c.) 7, 4, 1, ___________, -5, -8
d.) 64, 16, 4, ___________, 4 , 16
1
1
2. Consider the arithmetic sequence whose first three terms are given by: 4, 14, 24
a.) Determine the fourth, fifth and sixth terms of this sequence.
b.) Write an explicit formula for 𝑎𝑛 .
c.) Find 𝑎2015 .
3. For each of the following arithmetic sequences:
i. Write an explicit formula for the nth term in the sequence.
ii. Find the 2015th term in the sequence.
a.) 13, 25, 37, 49, 61, ...
b.) 24, 20, 16, 12, 8 ...
c.) -1, -2.5, -4, -5.5, …
d.) 100, 1100, 2100, 3100, …
Some times we will need to find the nth term of a sequence given two arbitrary terms in the
sequence. The key thing we must remember is that arithmetic sequences are just linear
functions.
a.) The third term in an arithmetic sequence is 10 and the fifth term is 26. If the first term is 𝑎1 ,
determine an equation for 𝑎𝑛 , the nth term in the sequence.
b.) In an arithmetic sequence, 𝑓(4) = 19 and 𝑓(7) = 31. Determine a formula for 𝑓(𝑛), the nth
term of this sequence.
c) An arithmetic sequence has a first term of 10 and a sixth term of 40. What is the 20th term of
this sequence?
Name: ___________________________________
Date: __________________
Arithmetic Sequences Homework
1. For each of the following arithmetic sequences:
i. State the common difference, d.
ii. Write an explicit formula for the nth term in the sequence.
iii. Find the 1234th term in the sequence.
a.) 10, 12, 14, 16, ...
b.) 30, 23, 16, 9, ...
c.)
n
1
2
3
4
5
f(n)
4
14
24
34
44
2. Write a formula for 𝑎𝑛 , the nth term, of the arithmetic sequence given that 𝑎6 = 15 and 𝑎10 =
63.
3. An auditorium has 21 rows of seats. The first row has 18 seats, and each succeeding row
has two more seats than the previous row.
a.) Write an explicit formula for the sequence.
b.) Determine how many seats are in the 21st row.
4. Below is a graph of the linear function a(n). Sketch a graph of an arithmetic sequence
that would correspond to a(n). (feel free to use the blank space below)
Write an explicit rule for the sequence a(n).
What are some similarities and differences between linear functions and arithmetic sequences?
Make sure to talk about the common difference.