No. of
Squares
No. of
Sticks
ARITHMETIC
SEQUENCE
LESSON GOALS:
1. Determine whether the given sequence is
arithmetic or not;
2. Derive the formula for AS;
3. Find an for each arithmetic sequence; and
4. Answer problem involving AS.
ARITHMETIC
OR NOT
Arithmetic Sequence
• An arithmetic sequence is a sequence in
which the difference between any two
consecutive terms is the same.
{21, 31, 41, 51…}
Arithmetic Sequence
• This constant difference is called the
common difference, and will be denoted by
d.
{21, 31, 41, 51…}
To find the common difference (d),
subtract any term from one that
follows it.
a1
a2
a3
a4
a5
2
5
8
11
14
3
3
3
3
Arithmetic Sequence
Observe these examples:
Arithmetic Sequence
7, 14, 21, 28…
28, 24, 20, 16, …
1 2 4
0, , , 1,
3 3 3
Common Difference
Arithmetic Sequence
Observe these examples:
Arithmetic Sequence
Common Difference
7, 14, 21, 28…
7
28, 24, 20, 16, …
1 2 4
0, , , 1,
3 3 3
Arithmetic Sequence
Observe these examples:
Arithmetic Sequence
Common Difference
7, 14, 21, 28…
7
28, 24, 20, 16, …
1 2 4
0, , , 1,
3 3 3
–4
Arithmetic Sequence
Observe these examples:
Arithmetic Sequence
Common Difference
7, 14, 21, 28…
7
28, 24, 20, 16, …
1 2 4
0, , , 1,
3 3 3
–4
1
3
BE CAREFUL: ALWAYS
CHECK TO MAKE SURE
THE DIFFERENCE IS
THE SAME BETWEEN
EACH TERMS !
Example 1:
Find the first term and the common
difference of each of these arithmetic
sequences.
a) 1, -4, -9, -14, ….
b) 11, 23, 35, 47, ….
Example 1:
Find the first term and the common difference of
each of these arithmetic sequences.
a) 1, -4, -9, -14, ….
Example 1:
Find the first term and the common difference of
each of these arithmetic sequences.
b) 11, 23, 35, 47, ….
Determine whether the given sequence is arithmetic or not.
If it is arithmetic, find the common difference and the next
two terms.
____ 1. 25, 21, 17, 13, 9, …
____ 2. 40, 20, 10, 5, …
____ 3. 6, 10, 14, 18, 22, …
____ 4. 1, 3, 9, 27, 81, …
____ 5. –1, 4, 9, 14, 19, …
List the next four terms of an arithmetic sequence, given a1
and d.
1. a1 = -4;
d=4
-4, ______, ______, ______, ______, …
2. a1 = 7; d = -3
7, ______, ______, ______, ______, …
3. a1 = 11; d = 8
11, ______, ______, ______, ______, …
4. a1 = 16;
16, ______, ______, ______, ______, …
d = -5
5. a1 = 2x + 1; d = x – 1
2x+ 1, ________, ________, ________, ________, …
List the next four terms of an arithmetic sequence, given a1
and d.
1. a1 = -4;
d=4
2. a1 = 7; d = -3
-4, ______, ______, ______, ______, …
7, ______, ______, ______, ______, …
List the next four terms of an arithmetic sequence, given a1
and d.
3. a1 = 11; d = 8
11, ______, ______, ______, ______, …
4. a1 = 16;
16, ______, ______, ______, ______, …
d = -5
List the next four terms of an arithmetic sequence, given a1
and d.
5. a1 = 2x + 1; d = x – 1
2x+ 1, ________, ________, ________, ________, …
Thank You!
LESSON GOALS:
1. Determine whether the given sequence is
arithmetic or not;
2. Derive the formula for AS;
3. Find an for each arithmetic sequence; and
4. Answer problem involving AS.
TH
N
FORMULA FOR THE
TERM OF AN
ARITHMETIC SEQUENCE
Formula for the nth term of an Arithmetic Sequence
a1
a2 = a1 + d
a3 = a1 + d + d = a1 + 2d
a4 = a1 + d + d + d = a1 + 3d
a5 = a1 + d + d + d + d = a1 + 4d
Formula for the nth term of an Arithmetic Sequence
a1 = a1 + 0d
a2 = a1 + 1d
a
=
a
+
(n
–
1)d
1
a3 = a1 + 2d n
a4 = a1 + 3d
a5 = a1 + 4d
Example 2:
Write a formula for the nth term
of the given arithmetic sequence.
a. 12, 19, 26, 33, 40, …
b. b. 9, 1, −7, −15, −23, …
Example 2:
Write a formula for the nth term of the given arithmetic
sequence.
an = a1 + (n
a. 12, 19, 26, 33, 40, …
Given:
a1 = 12
d=7
Solution:
an = a1 + (n – 1)d
an = 12 + (n – 1)7
an = 12 + 7n – 7
an = 5+7n
– 1)d
Answer:
an = 7n + 5
Example 2:
Write a formula for the nth term of the given arithmetic
sequence.
an = a1 + (n
b. 9, 1, −7, −15, −23, …
Given:
a1 = 9
d = –8
Solution:
an = a1 + (n – 1)d
an = 9 + (n – 1)–8
an = 9 + (-8n + 8)
an = 9 –8n + 8
an = 17 –8n
– 1)d
Answer:
an = –8n + 17
LESSON GOALS:
1. Determine whether the given sequence is
arithmetic or not;
2. Derive the formula for AS;
3. Find an for each arithmetic sequence; and
4. Answer problem involving AS.
TH
N
FINDING THE
TERM
OF AN ARITHMETIC
SEQUENCE
Example 3:
1. Find the 𝑎40 of the sequence 9, 15, 21,
27, …
2. Find the 15th term of the arithmetic
sequence 18, 22, 26, 30, 34, …
3. 2, 5, 8, … the 8th term
Example 3:
1.
an = a1 + (n – 1)d
Find the 𝑎40 of the sequence 9, 15, 21, 27, …
Given:
a1 = 9
d=6
n = 40
a40 = ?
Solution:
an = a1 + (n – 1)d
a40 = 9 + (40 – 1)6
a40 = 9 + (39)6
a40 = 9 + 234
a40 = 243
Answer:
a40 = 243
Example 3:
2.
an = a1 + (n – 1)d
Find the 15th term of the arithmetic sequence 18, 22, 26, 30, 34, …
Given:
a1 = 18
d=4
n = 15
a15 = ?
Solution:
an = a1 + (n – 1)d
a15 = 18 + (15 – 1)4
a15 = 18 + (14)4
a15 = 18 + 56
a15 = 74
Answer:
a15 = 74
Example 3:
3.
an = a1 + (n – 1)d
2, 5, 8, … the 8th term
Given:
a1 = 2
d=3
n=8
a8 = ?
Solution:
an = a1 + (n – 1)d
a8 = 2 + (8 – 1)3
a8 = 2 + (7)3
a8 = 2 + 21
a8 = 23
Answer:
a8 = 23
LESSON GOALS:
1. Determine whether the given sequence is
arithmetic or not;
2. Derive the formula for AS;
3. Find an for each arithmetic sequence; and
4. Answer problem involving AS.
ANSWER PROBLEM
INVOLVING ARITHMETIC
SEQUENCE
Example 4:
1. In the arithmetic sequence 8, 14, 20, 26,
32, … , which term is 122?
2. In the arithmetic sequence 4, 7, 10, 13,
…, which term has a value of 301?
3. In the arithmetic sequence 106, 102, 98,
94, …, which term has a value of 2?
Example 4:
1.
an = a1 + (n – 1)d
In the arithmetic sequence 8, 14, 20, 26, 32, … , which term is 122?
Given:
a1 = 8
d=6
n=?
an = 122
Solution:
120
6n
=
6
6
an = a1 + (n – 1)d
122 = 8 + (n – 1)6
𝑛
=
20
122 = 8 + (6n – 6)
122 = 8 + 6n – 6
Answer:
122 = 6n + 2
122 is the 20th term
120 = 6n
Example 4:
2.
an = a1 + (n – 1)d
In the arithmetic sequence 4, 7, 10, 13, …, which term has a value of
301?
Given:
a1 = 4
d=3
n=?
an = 301
Solution:
300
3n
=
3
3
an = a1 + (n – 1)d
301 = 4 + (n – 1)3
𝑛
=
100
301 = 4 + (3n – 3)
301 = 4 + 3n – 3
Answer:
301 = 3n + 1
301 is the 100th term
300 = 3n
Example 4:
3.
an = a1 + (n – 1)d
In the arithmetic sequence 106, 102, 98, 94, …, which term has a
value of 2?
Given:
a1 = 106
d = –4
n=?
an = 2
Solution:
–108
–4n
=
–4
–4
an = a1 + (n – 1)d
2 = 106 + (n – 1)–4
𝑛
=
27
2 = 106 + (–4n + 4)
2 = 106 – 4n + 4
Answer:
2 = 110 – 4n
2 is the 27th term
–108 = –4n
Example 5:
th
18
1. Find the
term of the arithmetic
sequence whose first term is 11 and
whose seventh term is 59.
rd
2. The 3 term of an arithmetic sequence
is 8 and the 16th term is 47. Find the
st
71 term.
3. Given a8 = 5 and a21 = –60, find a17.
Example 5:
an = a1 + (n – 1)d
Find the 18th term of the arithmetic sequence whose first term is 11
and whose seventh term is 59.
Solution: (solve for d )
Given:
Solution: (solve for a18 )
Given:
a1 = 11 an = a1 + (n – 1)d
a1 = 11 an = a1 + (n – 1)d
a7 = 59 59 = 11 + (7 – 1)d
n = 18 a18 = 11 + (18 – 1)8
59 = 11 + (6)d
n=7
a18 = 11 + (17)8
d=8
59 = 11 + 6d
d=?
a18 = ? a18 = 11 + 136
48 = 6d
a18 = 147
48
6d
=
6
6
Answer:
d=8
1.
a18 = 147
Example 5:
2.
an = a1 + (n – 1)d
The 3rd term of an arithmetic sequence is 8 and the 16th term is 47.
Find the 71st term.
Given:
a3/ak = 8
a16/an = 47
k=3
n = 16
d=?
a1 = ?
Solution: (solve for d)
𝑎𝑛 − 𝑎𝑘
𝑑=
𝑛−𝑘
47 − 8
𝑑=
16 − 3
39
𝑑=
13
𝒅=𝟑
Solution: (solve for a1 )
Let’s assign a3 as an
an = a1 + (n – 1)d
8 = a1 + (3 – 1)3
8 = a1 + (2)3
8 = a1 + 6
a1 = 2
Example 5:
2.
an = a1 + (n – 1)d
The 3rd term of an arithmetic sequence is 8 and the 16th term is 47.
Find the 71st term.
Given:
a1 = 2
d=3
n = 71
a71 = ?
Solution: (solve for a71 )
an = a1 + (n – 1)d
a71 = 2 + (71 – 1)3
a71 = 2 + (70)3
a71 = 2 + 210
a71 = 212
Answer:
a71 = 212
Example 5:
3.
an = a1 + (n – 1)d
Given a8 = 5 and a21 = –60, find a17.
Given:
a8/ak = 5
a21/an = –60
k=8
n = 21
d=?
a1 = ?
Solution: (solve for d)
𝑎𝑛 − 𝑎𝑘
𝑑=
𝑛−𝑘
−60 − 5
𝑑=
21 − 8
−65
𝑑=
13
𝒅 = −𝟓
Solution: (solve for a1 )
Let’s assign a8 as an
an = a1 + (n – 1)d
5 = a1 + (8 – 1)–5
5 = a1 + (7) –5
5 = a1 + (–35)
5 = a1 –35
40 = a1
a1 = 40
Example 5:
3.
an = a1 + (n – 1)d
Given a8 = 5 and a21 = –60, find a17.
Given:
a1 = 40
d = –5
n = 17
a17 = ?
Solution: (solve for a71 )
an = a1 + (n – 1)d
a17 = 40 + (17 – 1)–5
a17 = 40 + (16) –5
a17 = 40 + (–80)
a17 = 40 – 80
a17 = –40
Answer:
a71 = -40
Practice Activity
1. Find the nth term of a sequence 75, 68, 61, 54, …
2. How many terms are there in a sequence -8, -5, 2, 1, 4, …91?
3. The fourth term of an arithmetic sequence is 22.
if the ninth term is 62, what is the first term and
the common difference of the sequence?
Practice Activity
1.
an = a1 + (n – 1)d
Find the nth term of a sequence 75, 68, 61, 54, …
Answer:
Given: Solution:
an = –7n + 82
a1 = 75 an = a1 + (n – 1)d
d=7
an = 75 + (n – 1)–7
an = 75 + (– 7n + 7)
an = 75 – 7n + 7
an = –7n + 82
Practice Activity
2.
an = a1 + (n – 1)d
How many terms are there in a sequence -8, -5, -2, 1, 4, …91?
Given:
a1 = -8
d=3
n=?
an = 91
Solution:
102
3n
=
3
3
an = a1 + (n – 1)d
91 = -8 + (n – 1)3
𝑛
=
34
91 = -8 + (3n – 3)
91 = -8 + 3n – 3
Answer:
91 = 3n – 11
There are 34 terms
102 = 3n
Practice Activity
3.
an = a1 + (n – 1)d
The fourth term of an arithmetic sequence is 22. if the ninth term is
62, what is the first term and the common difference of the sequence?
Given:
a4/ak = 22
a9/an = 62
k=4
n=9
d=?
a1 = ?
Solution: (solve for d)
𝑎𝑛 − 𝑎𝑘
𝑑=
𝑛−𝑘
62 − 22
𝑑=
9−4
40
𝑑=
5
𝒅=𝟖
Solution: (solve for a1 )
Let’s assign a4 as an
an = a1 + (n – 1)d
22 = a1 + (4 – 1)8
22 = a1 + (3)8
22 = a1 + 24
-2 = a1
a1 = -2
Thank You!