Lesson 3-5
Example 1 Identify Arithmetic Sequences
Determine whether each sequence is an arithmetic sequence. Explain.
a. 0.7, 0.5, 0.3, 0.1,…
0.7, 0.5, 0.3, 0.1
-0.2 -0.2 -0.2
The difference between each term in the sequence is constant. Therefore, this
sequence is arithmetic.
1 1 1 1
b. - , - , - , - …
3 9 27 81
1 1 1 1
- - - -
3 9 27 81
2 2 2
+ + +
9 27 81
This is not an arithmetic sequence because the difference between terms is not constant.
Example 2 Find the Next Term
Find the next three terms of the arithmetic sequence 36, 39, 42, 45, … .
Step 1 Find the common difference by subtracting successive terms.
36 39 42 45
+3 +3 +3 The common difference is 3.
Step 2 Add 3 to the last term of the sequence to get the next term.
45 48 51 54
+3 +3 +3
The next three terms in the sequence are 48, 51, and 54.
Example 3 Find the nth Term
a. Write an equation for the nth term of the arithmetic sequence 2, -2, -6, -10, … .
Step 1 Find the common difference.
2 -2 -6 -10
-4 -4 -4 The common difference is –4.
Step 2 Write an equation.
a n = a 1 + (n – 1)d Formula for the nth term
a n = 2 + (n – 1)(-4) a1 = 2, d = -4
a n = 2 + (-4n) + 4 Distributive Property
a n = -4n + 6 Simplify.
b. Find the 25th term of the sequence.
Substitute 25 for n in the formula for the nth term.
a n = -4n + 6 Formula for the nth term
a 25 = -4(25) + 6 Replace n with 25.
a 25 = -100 + 6 Multiply.
a 25 = -94 Simplify.
c. Graph the first five terms of the sequence.
n -4n + 6 an (n, an)
1 -4(1) + 6 2 (1, 2)
2 -4(2) + 6 -2 (2, -2)
3 -4(3) + 6 -6 (3, -6)
4 -4(4) + 6 -10 (4, -10)
5 -4(5) + 6 -14 (5, -14)
d. Which term of the sequence is –30?
In the formula for the nth term, substitute –30 for an.
an = –4n + 6 Formula for nth term
–30 = –4n + 6 an = –30
–30 – 6 = –4n + 6 – 6 Subtract 6 from each side.
–36 = –4n Simplify.
9=n Divide each side by –4.
So, –30 is the 9th term of the sequence.
Real-World Example 4 Arithmetic Sequences as Functions
TRAMPOLINES Brianna is jumping on a trampoline. The arithmetic
sequence 2.4, 4.8, 7.2, 9.6, … represents the time in seconds in which her
feet land on the trampoline.
a. Write a function to represent this sequence.
The first term, a1 is 2.4. Find the common difference.
2.4 4.8 7.2 9.6
+2.4 +2.4 +2.4
The common difference is 2.4.
an = a1 + (n – 1)d Formula for the nth term
= 2.4 + (n – 1)2.4 a1 = 2.4 and d = 2.4
= 2.4 + 2.4n – 2.4 Distributive Property
= 2.4n Simplify.
The function is a(n) = 2.4n.
b. Graph the function and determine the domain.
The rate of change of the function is 2.4. Make a table and plot points.
n a(n) Jumping on a
Trampoline
1 2.4
Number of Seconds
2 4.8 14
3 7.2 12
4 9.6 10
5 12.0 8
6
4
2
O
1 2 3 4 5 6 7
Times Landing
on Trampoline
The domain of the function is the number of times her feet land on the
trampoline. So, the domain is {1, 2, 3, 4, … }.