Arithmetic Sequences
Arithmetic Sequence
• a sequence where consecutive terms present a common difference. I.e. each term, after the first,
is formed by adding a fixed amount (constant) to the preceeding term.
The sequence, {8, 3, -2, -7, -12}, is an arithmetic sequence because starting at t1 , you
add -5 to each consecutive term. This constant, -5, is called the common difference of
the first level. We denote this by D1 = −5 .
{8,
•
3,
-2,
-7,
-12}
t 2 − t1
t3 − t 2
t 4 − t3
t5 − t 4
= 3−8
= −2 − 3
= −7 − − 2
= −12− − 7
= −5
= −5
= −5
= −5
If the sequence is graphed (term number vs value of term) it creates a linear function.
The graph for {8, 3, -2, -7, -12} is below.
•
Formula for finding the general term/equation/rule/nth term of arithmetic sequences:
t n = t1 + (n − 1)d
nth
term
first
term
term
number
common
difference
Example 1:
Determine if the following sequence is arithmetic.
{4, 8, 12, 16, 20, 24, 28}
Solution:
__________________________________________________________________________________
Example 2:
Determine if the following sequence is arithmetic.
{10, 18, 26, 34, 40, 48, 56}
Solution:
__________________________________________________________________________________
Example 3:
Solution:
Find the nth term of the sequence {3, 9, 15, 21, …} and then find term number 80.
Example 4:
Find the equation that would represent the sequence {6, 8, 10, 12} if graphed.
Solution:
NOTE: WHAT DO YOU NOTICE ABOUT THE SLOPE OF THE EQUATION AND THE FIRST
LEVEL COMMON DIFFERENCE?
Arithmetic Sequences
Fill in the chart below. Find the common difference and t200 ONLY for all arithmetic sequences.
#
Sequence
1.
{-2, -7, -12, -17, …}
2.
{2, 4, 6, 8, …}
3.
{1, 5, 9, 13, …}
4.
{3, 4, 6, 10, 18, …}
5.
{4.2, 6.2, 8.2, 10.2,…}
6.
{1, 3, 6, 10, …}
7.
⎧1 2 3 4 ⎫
⎨ , , , ,...⎬
⎩3 5 7 9 ⎭
⎧1 2 1 ⎫
⎨ , ,1,1 ,...⎬
3 ⎭
⎩3 3
8.
9.
{100, 90, 80, 70, …}
10.
{-3, -9, -15, -21, …}
11.
{3, 6, 9, 12, …}
12.
{1, 6, 11, 16, …}
13.
{3, 6, 10, 15, …}
14.
{2.7, 6.7, 10.7,14.7,…}
15.
{1, 9, 17, 25, …}
16.
{200, 100, 0, -100, …}
17.
{10, 15, 20, 25, …}
18.
{10, 15, 25, 40, …}
19.
{1, 2, 3, 4, …}
20.
{0, -0.4, -0.8, -1.2, …}
Next Three
Terms
-22. -27, -32
Arithmetic?
(Yes/ No)
Yes
Common
Difference
-5
Relation
t200
tn = -2 + (n – 1)(-5)
tn = -5n + 3
-997
Arithmetic Sequences
1. A contractor is building a chain-link fence containing 63 sections. If each section
(link) is a square made from four metal rods, and any two adjacent sections
share one rod.
a. How many rods are needed for one link?
b. How many rods are needed for two links?
c. How many rods are needed for three links?
d. Create a sequence for 1 – 6 links.
e. In your sequence do you find a pattern? Explain.
f. Create a data table
term
1
2
number (n)
Value of
term (tn)
3
4
5
6
g. Subtract the first term from the second term, the second from the third and
so on… to find the common difference. (t2 – t1, t3 – t2 … tn – tn-1)
h. What type of sequence is this?
i. Find the rule that represents the sequence.
j. Use your pattern to find how many rods will be needed to build the whole
fence of 63 links.
k. Graph your sequence (value of term (tn) vs. term number (n)) and state
what type of function it looks like.
2. A contractor is building yet another chain-link fence containing 63 sections. If
each section (link) is a square made from six metal rods, and any two adjacent
sections share one rod.
a. How many rods are needed for one link?
b. How many rods are needed for two links?
c. How many rods are needed for three links?
d. Create a sequence for 1 – 6 links.
e. In your sequence do you find a pattern? Explain.
f. Create a data table
term
1
number (n)
2
3
4
5
6
term (tn)
g. Subtract the first term from the second term, the second from the third and
so on… to find the common difference. (t2 – t1, t3 – t2 … tn – tn-1)
h. What type of sequence is this?
i. Find the nth term of the sequence.
j. How many rods will be needed to build the whole fence of 63 links.
k. Graph your sequence (value of term (tn) vs. term number (n)) and state
what type of function it looks like.
3. A bridge railing is to be formed by connecting sections of equilateral triangles.
The contractor wants to find the number of rods represented by tn, the nth term of
the sequence. { 3, 5, 7 …, tn}
a. Explain why the terms t1 to t5 in the sequence can be expressed as
follows:
t1 = 3 + 0 x 2
t2 = 3 + 1 x 2
t3 = 3 + 2 x 2
t4 = 3 + 3 x 2
t5 = 3 + 4 x 2
b. Express the values of t6 to t10, in a similar manner.
c. Explain how to express tn using a rule.
d. How many metal rods are needed for a bridge railing constructed from 200
equilateral triangles?
4. A decorative railing is to be constructed from sections in the shape of a regular
hexagon.
a. Create a sequence for t1 to t10 representing the number of metal rods
needed to build from one to ten sections.
b. Find the nth term of the sequence representing the number of metal rods
needed to construct a railing using n hexagonal sections.
c. How many metal rods are needed to construct a railing using 200 regular
hexagons?
5. Determine if the following sequences are arithmetic. If it is an arithmetic
sequence, find the nth term.
(a) { -4, -8, -12, -16, …}
(b) { 2, 7, 12, 17, …}
(c) { 1, 3, 9, 27, 81, 243, …}
(d) { 6.5, 8.5, 10.5, 12.5, 14.5, …}
(e) { 1, 4, 9, 16, 25, 36, …}
(f) { 1 , 3 , 1, 5 , 3 , …}
2
4
4
2
(g) { 24, 19, 14, 9, …}
6. Create an arithmetic sequence in which each term of the sequence of differences
is
a. –5
b. –3.5
c. – ¾
d. –150
7. In each pattern below, four arrangements of dots are shown. For each pattern,
write the number of dots in the arrangements as the first four terms in a
sequence. If the sequence is arithmetic, explain why it is arithmetic and find the
nth term.
a.
b.
c.
8. The function t n = 3n + 1 generates the sequence {4, 7, 10, 13, 16, …}. The values
of the function can be represented in a scatterplot as shown
a. Explain how you know the
graph represents an arithmetic
sequence?
Number of Rods
Arithmetic Sequence
18
16
14
12
10
8
6
4
2
0
b. If the points are joined, what is
the slope of the graph?
0
1
2
3
4
5
6
Term Number
c. What is the common difference of the sequence?
d. How does the slope and common difference relate?
9. Use each function below to generate the first six terms of a sequence.
tn = -3n + 4.5
tn = -3n – 0.5
tn = -3n + 2
a. How do you know each is an arithmetic sequence?
b. Graph each function. Why is each graph a straight line?
c. Why is each line parallel to the other lines?
d. How does the common difference appear to be related to the slope of the
graph of each function used to generate an arithmetic sequence?
10. What is the common difference between the successive terms in the sequence
1
1
generated by t n = − n + ?
3
2
11. Complete each sequence and find the nth term.
a. 2, 4, 6, 8, 10, ___, ___, ___, …, nth
b. 3, 6, 9, 12, ___, ___, ___, …, nth
c. 17, 12, 7, 2, ___, ___, ___, …, nth
12. Explain why the sequences in #11 are called arithmetic sequences.
Value of term
13. If this graph represents a sequence of numbers
a. Is the sequence arithmetic?
18
16
14
12
10
8
6
4
2
0
b. What would be the value of
the 8th term?
c. Describe the nth term.
0
1
2
3
4
5
Term Number
d. Write the function for the graph/sequence.
Answers:
1. a)
b)
c)
d)
e)
f)
g)
h)
i)
4
7
10
{4, 7, 10, 13, 16, 19}
add three to each term
D1 = 3
arithmetic
tn=3(n-1) + 4
tn = 3n + 1
j) 190
k) linear function
2. a)
b)
c)
d)
e)
f)
g)
h)
i)
6
11
16
{6, 11, 16, 21, 26, 31}
add five to each term
D1 = 5
arithmetic
tn = 5(n-1) + 6
tn = 5n + 1
j) 316
k) linear function
3. a) because an arithmetic
sequence
b) t6 = 3 + 5 x 2, t7 = 3 + 6 x 2,
t8 = 3 + 7 x 2, t9 = 3 + 8 x 2,
t10 = 3 + 9 x 2
c) tn = 2(n-1) + 3
tn = 2n + 1
d) 401
4. a) {6, 11, 16, 21, 26, 31}
b) tn = 5(n-1) + 6
tn = 5n + 1
c) 1001
5. a) Yes, tn = -4(n-1) – 4
tn = -4n
b) Yes, tn = 5(n-1) + 2
tn = 5n – 3
c) No
d) Yes, tn = 2(n-1) + 6.5
tn = 2n + 4.5
e) No
f) Yes, tn = ¼ (n-1) + ½
tn = ¼ n + ¼
g) Yes, tn = -5(n-1) + 24
tn = -5n + 29
6. answers will vary
7. a) No
b) No
c) Yes, tn = 2n + 1
8. a)
b)
c)
d)
D1 = 3
m=3
D1 = 3
same
9. a)
b)
c)
d)
D1 = -3
linear functions
the slope is the same
same
10. a) −
1
3
11. a) 12, 14, 16
tn = 2n
b) 15, 18, 21
tn = 3n
c) -3, -8, -13
tn = -5n + 22
12. 1st level common differences
13. a)
b)
d)
e)
Yes
-12
tn = -4n + 20
tn = -4n + 20