Five-Minute Check
CCSS
Then/Now
New Vocabulary
Key Concept: nth Term of an Arithmetic Sequence
Example 1: Find the nth term
Example 2: Write Equations for the nth Term
Example 3: Find Arithmetic Means
Key Concept: Partial Sum of an Arithmetic Series
Example 4: Use the Sum Formulas
Example 5: Find the First Three Terms
Key Concept: Sigma Notation
Example 6: Standardized Test Example: Use Sigma Notation
Over Lesson 10–1
Determine whether the sequence is arithmetic,
geometric, or neither.
18, 11, 4, …
A.arithmetic
B.geometric
C.neither
Over Lesson 10–1
Determine whether the sequence is arithmetic,
geometric, or neither.
18, 11, 4, …
A.arithmetic
B.geometric
C.neither
Over Lesson 10–1
Determine whether the sequence is arithmetic,
geometric, or neither.
1, –2, 4, –8, …
A.arithmetic
B.geometric
C.neither
Over Lesson 10–1
Determine whether the sequence is arithmetic,
geometric, or neither.
1, –2, 4, –8, …
A.arithmetic
B.geometric
C.neither
Over Lesson 10–1
Determine whether the sequence is arithmetic,
geometric, or neither.
5, 6, 8, 11, …
A.arithmetic
B.geometric
C.neither
Over Lesson 10–1
Determine whether the sequence is arithmetic,
geometric, or neither.
5, 6, 8, 11, …
A.arithmetic
B.geometric
C.neither
Over Lesson 10–1
Find the next three terms of the sequence.
25, 50, 75, 100, …
A.125, 150, 175
B.125, 250, 500
C.125, 145, 175
D.150, 200, 225
Over Lesson 10–1
Find the next three terms of the sequence.
25, 50, 75, 100, …
A.125, 150, 175
B.125, 250, 500
C.125, 145, 175
D.150, 200, 225
Over Lesson 10–1
Find the next three terms of the sequence.
–1, –6, –36, –216, …
A.–236, –266, –336
B.–306, –336, –416
C.–1296, –7776, –46,656
D.–1296, –3888, –11,664
Over Lesson 10–1
Find the next three terms of the sequence.
–1, –6, –36, –216, …
A.–236, –266, –336
B.–306, –336, –416
C.–1296, –7776, –46,656
D.–1296, –3888, –11,664
Over Lesson 10–1
Find the first term and the ninth term of the
arithmetic sequence.
___, 4.5, 7, 9.5, 12, …
A.2;14.5
B.2.5;22
C.2;22
D.2.5;14.5
Over Lesson 10–1
Find the first term and the ninth term of the
arithmetic sequence.
___, 4.5, 7, 9.5, 12, …
A.2;14.5
B.2.5;22
C.2;22
D.2.5;14.5
Content Standards
A.CED.4 Rearrange formul
as to highl
ight a
quantityofinterest,using the same
reasoning as in sol
ving equations.
Mathematical Practices
8 Look forand express regul
arityin repeated
reasoning.
You determined whethera sequence was
arithmetic.
• Use arithmetic sequences.
• Find sums ofarithmetic series.
• arithmetic means
• series
• arithmetic series
• partialsum
• sigma notation
Find the nth Term
Find the 20th term of the arithmetic sequence
3, 10, 17, 24, … .
Step 1Find the common difference.
24 – 17 = 717 – 10 = 710 – 3 = 7
So,d = 7.
Find the nth Term
Step 2Find the 20th term.
an=a1 + (n – 1)dnth term ofan
arithmetic sequence
a20=3 + (20 – 1)7a1 = 3,d = 7,n = 20
=3 + 133 or136Simpl
ify.
Answer:
Find the nth Term
Step 2Find the 20th term.
an=a1 + (n – 1)dnth term ofan
arithmetic sequence
a20=3 + (20 – 1)7a1 = 3,d = 7,n = 20
=3 + 133 or136Simpl
ify.
Answer: The 20th term ofthe sequence is 136.
Find the 17th term of the arithmetic sequence 6, 14,
22, 30, … .
A.134
B.140
C.146
D.152
Find the 17th term of the arithmetic sequence 6, 14,
22, 30, … .
A.134
B.140
C.146
D.152
Write Equations for the nth Term
A. Write an equation for the nth term of the
arithmetic sequence below.
–8, –6, –4, …
d=–6 – (–8)or2;–8 is the first term.
an=a1 + (n – 1)dnth term ofan arithmetic sequence
an=–8 + (n – 1)2a1 = –8 and d = 2
an=–8 + (2n – 2)Distributive Property
an=2n – 10Simpl
ify.
Answer:
Write Equations for the nth Term
A. Write an equation for the nth term of the
arithmetic sequence below.
–8, –6, –4, …
d=–6 – (–8)or2;–8 is the first term.
an=a1 + (n – 1)dnth term ofan arithmetic sequence
an=–8 + (n – 1)2a1 = –8 and d = 2
an=–8 + (2n – 2)Distributive Property
an=2n – 10Simpl
ify.
Answer: an = 2n – 10
Write Equations for the nth Term
B. Write an equation for the nth term of the
arithmetic sequence below.
a6 = 11, d = –11
First,find a1.
an=a1 + (n – 1)dnth term ofan arithmetic sequence
11=a1 + (6 – 1)(–11)a6 = 11,n = 6,and d = –11
11=a1 – 55Mul
tipl
y.
66=a1Add 55 to each side.
Write Equations for the nth Term
Then write the equation.
an=a1 + (n – 1)dnth term ofan arithmetic sequence
an=66 + (n – 1)(–11)a1 = 66,and d = –11
an=66 + (–11n + 11)Distributive Property
an=–11n + 77Simpl
ify.
Answer:
Write Equations for the nth Term
Then write the equation.
an=a1 + (n – 1)dnth term ofan arithmetic sequence
an=66 + (n – 1)(–11)a1 = 66,and d = –11
an=66 + (–11n + 11)Distributive Property
an=–11n + 77Simpl
ify.
Answer: an = –11n + 77
A. Write an equation for the nth term of the
arithmetic sequence below.
–12, –3, 6, …
A.an = –9n – 21
B.an = 9n – 21
C.an = 9n + 21
D.an = –9n + 21
A. Write an equation for the nth term of the
arithmetic sequence below.
–12, –3, 6, …
A.an = –9n – 21
B.an = 9n – 21
C.an = 9n + 21
D.an = –9n + 21
B. Write an equation for the nth term of the
arithmetic sequence below.
a4 = 45, d = 5
A.an = 5n + 25
B.an = 5n – 20
C.an = 5n + 40
D.an = 5n + 30
B. Write an equation for the nth term of the
arithmetic sequence below.
a4 = 45, d = 5
A.an = 5n + 25
B.an = 5n – 20
C.an = 5n + 40
D.an = 5n + 30
Find Arithmetic Means
Find the arithmetic means in the sequence
21, ___, ___, ___, 45, … .
Step 1Since there are three terms between the first
and l
ast terms given,there are 3 + 2 or5
totalterms,so n = 5.
Step 2Find d.
an=a1 + (n – 1)dFormul
a forthe nth term
45=21 + (5 – 1)dn = 5,a1 = 21,a5 = 45
45=21 + 4d Distributive Property
24=4dSubtract 21 from each side.
6=dDivide each side by4.
Find Arithmetic Means
Step 3Use the val
ue ofd to find the three
arithmetic means.
21 27333945
+6
Answer:
+6
+6
+6
Find Arithmetic Means
Step 3Use the val
ue ofd to find the three
arithmetic means.
21 27333945
+6
+6
+6
+6
Answer: The arithmetic means are 27,33,and 39.
Find the three arithmetic means between
13 and 25.
A.16, 19, 22
B.17, 21, 25
C.13, 17, 21
D.15, 17, 19
Find the three arithmetic means between
13 and 25.
A.16, 19, 22
B.17, 21, 25
C.13, 17, 21
D.15, 17, 19
Use the Sum Formulas
Find the sum 8 + 12 + 16 + … + 80.
Step 1a1 = 8,an = 80,and d = 12 – 8 or4.
W e need to find n before we can use one of
the formul
as.
an=a1 + (n – 1)dnth term ofan
arithmetic sequence
80=8 + (n – 1)(4)an = 80,a1 = 8,
and d = 4
80=4n + 4Simpl
ify.
19=nSol
ve forn.
Use the Sum Formulas
Step 2Use eitherformul
a to find Sn.
Sum formul
a
a1 = 8,n = 19,
d =4
Simpl
ify.
Answer:
Use the Sum Formulas
Step 2Use eitherformul
a to find Sn.
Sum formul
a
a1 = 8,n = 19,
d =4
Simpl
ify.
Answer: 836
Find the sum 5 + 12 + 19 + … + 68.
A.318
B.327
C.340
D.365
Find the sum 5 + 12 + 19 + … + 68.
A.318
B.327
C.340
D.365
Find the First Three Terms
Find the first three terms of an arithmetic series in
which a1 = 14, an = 29, and Sn = 129.
Step 1Since you know a1,an,and Sn,use
to find n.
Sum formul
a
Sn = 129,a1 = 14,
an = 29
Simpl
ify.
Divide each side by43.
Find the First Three Terms
Step 2Find d.
sequence
an=a1 + (n – 1)dnth term ofan arithmetic
29=14 + (6 – 1)dan = 29,a1 = 14,n = 6
15=5dSubtract 14 from each
side.
3=dDivide each side by5.
Find the First Three Terms
Step 3Use d to determine a2 and a3.
a2=14 + 3 or17
a3=17 + 3 or20
Answer:
Find the First Three Terms
Step 3Use d to determine a2 and a3.
a2=14 + 3 or17
a3=17 + 3 or20
Answer: The first three terms are 14,17,and 20.
Find the first three terms of an arithmetic series in
which a1 = 11, an = 31, and Sn = 105.
A.16, 21, 26
B.11, 16, 21
C.11, 17, 23, 30
D.17, 23, 30, 36
Find the first three terms of an arithmetic series in
which a1 = 11, an = 31, and Sn = 105.
A.16, 21, 26
B.11, 16, 21
C.11, 17, 23, 30
D.17, 23, 30, 36
Use Sigma Notation
Evaluate
.
A.23B.70
C.98D.112
Read the Test Item
You need to find the sum ofthe series.
Find a1,an,and n.
Use Sigma Notation
Method 1Since the sum is an arithmetic series,use
the formul
a
terms.
a1 = 2(3)+ 1 or7,and a8 = 2(10)+ 1
or21
.There are 8
Use Sigma Notation
Solve the Test Item
Method 2Find the terms byrepl
acing k with
3,4,...,10.Then add.
Use Sigma Notation
Answer:
Use Sigma Notation
Answer: The sum ofthe series is 112.The correct
answeris D.
Evaluate
A.85
B.95
C.108
D.133
.
Evaluate
A.85
B.95
C.108
D.133
.