Sequences
A. Sequences
Each number in a sequence is called a term.
1. The term to term rule
The term to term rule can be used to find the next few terms
Write the term to term rule then find the next 3 terms
a.
2, 5, 8, 11, 14, ...
Add 3
17, 20, 23
b. 20, 18, 16, 14, 12, ...
Subtract 2
10, 8, 6
c.
Multiply by 2
64, 128, 256
1, 2, 4, 8, 16, 32, ...
The nth term is the rule that will calculate any term in a sequence
2. The nth term (Tn)
Using the nth term given write the first six terms in the sequence
4, 7, 10, 13, 16, 19
a. Tn = 3n + 1
Substitute the term position
number into the nth term
b.
Tn = 2n - 1
1, 3, 5, 7, 9, 11
c.
Tn = 30 – 2n
28, 26, 24, 22, 20, 18
3. Finding the nth term (Tn) for a linear or arithmetic sequence
Find the nth term
a. 3, 7, 11, 15, 19, …
4
4
4
4
3
3
3
3
Increasing by 4, so
the nth term has 4n
b. 5, 8, 11, 14, 17, …
c. 20, 18, 16, 14, 12, …
-2
-2
d)
-2
Decreasing by 2, so
the nth term has -2n
-2
Tn = 4n - 1
Tn = 3n + 2
Tn = -2n + 22 or 22 – 2n
The pattern below is made of matchsticks
(i)
Draw pattern 4
(ii)
What is the nth term for the number of sticks?
4, 7, 10, 13
(iii)
Tn = 3n + 1
Count the sticks in
each pattern to write
a sequence then find
the nth term
How many sticks will there be in pattern 25?
T25 = 3 x 25 + 1 = 75 + 1 = 76
B. Finding missing terms in an arithmetic sequence
An arithmetic sequence increases or decreases by a constant amount each time.
Find the missing terms
a) 2, a, b, 14
n
n
n
2 + 3n = 14
3n = 12
n = 12
3
n = 4
2, 6, 10, 14
a=6
b = 10
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Find the missing terms
b) 5, c, d, e, 33
n
n
n
n
Form an
equation then
solve it
5 + 4n = 33
4n = 28
n = 28
4
n = 7
5, 12, 19, 26, 33
c = 12
d = 19 e = 26
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Sequences
C. Geometric sequences
In a geometric sequence the ratio between each term is a multiplier
find the nth term
a. 2, 4, 8, 16, 32, …
x2
x2
x2
tn = 2n
x2
b. 1, 2, 4, 8, 16, …
x2
x2
x2
Tn = 2
x2
c. 0.8, 1.6, 3.2, 6.4, 12.8, …
x2
x2
x2
x2
Multiplying by 2, so the nth
term has a power of 2
n-1
Tn = 0.8 x 2n-1
20 =
21 =
22 =
23 =
1
2
4
8
D. Finding the nth term of a quadratic sequence
A quadratic sequence will contain an n2 term. We will need to find second differences
before we find the sequence changing by a constant amount.
The coefficient of the n2 term is half the value of the second differences.
find the nth term
a.
5, 11, 19, 29, 41, …
6
12
10
8
The second differences are 2, so 1n2
2
2
2
1,
4,
9, 16, 25 -
4,
7, 10, 13, 16
3
3
Find the nth term for the remaining sequence. It will be linear.
3
3
Subtract n2 from the original sequence
3n + 1
Tn = n2 + 3n + 1
b. 3, 12, 25, 42, 63, …
9
13
17
4
4
4
21
21
The second differences are 4, so 2n2
2,
8, 18,
32, 50 –
1,
4,
10, 13
3
7,
3
6, 13
7
26,
13
6
Find the nth term for the remaining sequence. It will be linear.
3
3
3n - 2
c.
Subtract 2n2 from the original sequence
Tn = 2n2 + 3n - 2
45, 70, …
19
6
6
25
21
The second differences are 6, so 3n2
3, 12, 27, 48,
75 –
Subtract 3n2 from the original sequence
3,
-5
Find the nth term for the remaining sequence. It will be linear.
1, -1,
-2
-2
-3,
-2
-2
-2n + 5
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Tn = 3n2 - 2n + 5
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Sequences
E. Using the nth term
1. We can use the nth term to determine the position of a particular term or to determine
whether a particular term will be in a sequence.
a. Find the value of the 50th term in the sequence 9, 15, 19, 23, 27 , …
4
4
4
4
tn = 4n + 5
th
T50 = 4 x 50 + 5
First find the n term
T50 = 205
b. Which term will have the value 293?
4n + 5 = 293
4n = 288
n=
c. Will 367 be in the sequence?
Form an equation
then solve it
𝟐𝟖𝟖
𝟒
4n + 5 = 367
4n = 362
n=
𝟑𝟔𝟐
𝟒
n = 90.5
n = 72
term 72
367 would be
between term
90 and 91
no 367 will not be in the sequence
2. We can find the value of missing terms.
The nth term of a sequence is 3n + 4. The sum of two consecutive terms is 221.
which terms are they and what is their value?
Let the two term positions be k and k + 1
tk = 3k + 4 tk + 1 = 3(k + 1) + 4
= 3k + 4 and 3k + 3 + 4
= 3k + 4 and 3k + 7
Substitute these positions into the nth term
to find an expression for their value
sum = 3k + 4 + 3k + 7 = 221
6k + 11 =221
6k = 210
k=
𝟐𝟏𝟎
Start by showing the term numbers
you are going to use
Find their sum and form an equation
Solve the equation
𝟔
k = 35
The positions are 35 and 36
Substitute these positions into the nth term
to find their value
3n + 4 = 3 x 35 + 4 = 105 + 4 = 109
3n + 4 = 3 x 36 + 4 = 108 + 4 = 112
The values are 109 and 112
Check the sum: 109 + 112 = 221
Just as we were told!
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