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SEQUENCES
SEQUENCE
Numbers that follow a pattern or a rule
Rule: it defines how successive terms are obtained, the rule can be recurrence or n-th term rule
Recurrence relation: previous terms to define the next term (term to term rule)
N-th term rule: uses the value of 𝑛
𝑎 = 2𝑛ଶ^ + 3𝑛 + 4
Knowledge of previous terms is required. Directly n-th term can be calculated. 2
TYPES OF SEQUENCE
Arithmetic Sequence: successive terms differ by same value
Geometric Sequence: successive terms are in same ratio
First term Common difference 3 9 27 81 243 × 3 × 3 × 3 × 3 First term Common ratio 3 TYPES OF SEQUENCE
Linear Sequence: successive terms differ by same value (also referred as arithmetic sequence)
Quadratic Sequence: second differences of successive terms is same
First term First difference +2 +2^ +2 (^) Second difference 4
PRACTICE
What is the term to term rule and the next two terms of the sequence:
A sequence has 𝑛-th term rule as 2𝑛 + 1. Find the first four terms.
Find the 𝑛-th term of the following sequence 3, 7, 11, 15, 19
Find the 𝑛-th term of the sequence −2, 5, 12, 19, 26
The sum of two consecutive terms in a sequence given by 𝑛-th term 3𝑛 + 8 is 109. Find the value of
these two terms.
A sequence has 𝑛-th term 4𝑛 + 1
Find the 12 -th term in sequence A term in the sequence is 77. Find the position of the term 7 8
GEOMETRIC SEQUENCE
General representation of geometric sequence
First term: 𝑎 Common ratio: 𝑟 N-th term: 𝑎
Rule of geometric sequence is:
Recurrence (term-to-term): 𝑎 = 𝑎ିଵ × 𝑟 n-th term: 𝑎 = 𝑎 × 𝑟ିଵ Example (Common ratio 𝒓 = 𝟒) General (Common ratio 𝒓) Term 1 𝑎 3 2 𝑎 × 𝑟 3 × 4 = 12 3 𝑎 × 𝑟ଶ 12 × 4 = 48 4 𝑎 × 𝑟ଷ 48 × 4 = 192 ⋮ n-th 𝑎 × 𝑟ିଵ 3 × 4ିଵ 9 EXAMPLE
What are the next three terms of a sequence that has a first term of 1 , where the term to term rule
is multiply by 2?
What is the 𝑛-th term rule?
𝑎 = 1 × 2ିଵ
What is the term-to-term rule?
𝑎 = 𝑎ିଵ × 2
Term 1 1 2 1 × 2 = 2 3 2 × 2 = 4 4 4 × 2 = 8 10
QUADRATIC SEQUENCE (FIRST-STEP)
𝑎 = 𝑎𝑛ଶ^ + 𝑏𝑛 + 𝑐
First step finds the coefficient 𝑎
𝑎 is second difference divided by 2
Now define 𝑆 𝑛 = 𝑎𝑛ଶ
+3 +5 +7 +9 First difference +2 +2^ +2 Second difference 𝑎 = 2 𝑆^ 𝑛^ =^ 𝑛ଶ 2
13 QUADRATIC SEQUENCE (SECOND-STEP)
Second step finds the coefficients 𝑏 and 𝑐
Tabulate the given sequence and 𝑆[𝑛]
Compute the difference of the two rows
If D is a sequence calculate the n-th term for
this sequence 𝐿[𝑛]
Combine 𝑆[𝑛] and 𝐿[𝑛]
𝑎 = 𝑆 𝑛 + 𝐿[𝑛]
Sequence 2 5 10 17 26 𝑆[𝑛] 1 4 9 16 25 Difference (D) 1 1 1 1 1 𝐿 𝑛 = 1 𝑎 = 𝑛ଶ^ + 1
14
QUADRATIC SEQUENCE
Deriving the 𝑛-th term of a quadratic sequence
𝑎 = 𝑎𝑛ଶ^ + 𝑏𝑛 + 𝑐
𝑎 is second difference divided by 2, define 𝑆[𝑛]
Tabulate to find D and 𝐿[𝑛]
Combine the linear sequence formula with 𝑛ଶ
term
+6 +8 +10 +12 First difference +2 +^2 + 2 Second difference 𝑎 = 1, 𝑆 𝑛 = 𝑛ଶ Sequence -1 5 13 23 35 𝑆[𝑛] 1 4 9 16 25 Difference (D) -2 1 4 7 10 𝐿 𝑛 = 3𝑛 − 5 𝑎 = 𝑛ଶ^ + 3𝑛 − 5
15 PRACTICE
Work out the 𝑛-th term of the sequence 5, 11, 21, 35
The 𝑛-th term of the sequence 9, 17, 27, 39
The n-th term of the sequence is 2 𝑛ଶ.
Find the 4-th term of the sequence. Is number 400 a term of the sequence. 16
PRACTICE
A sequence is defined by the formula 5𝑛 − 4
Find the first five terms of the sequence. Explain why 108 is not a term in the sequence.
The first five terms of an arithmetic progression are −3, 2, 7, 12, 17. Find the formula for the 𝑛-th
term of the sequence.
The first five terms of a sequence are 7, 11, 15, 19, 23.
Find the next two terms. Give the term-to-term rule of the sequence. 19 PRACTICE
The 𝑛-th term of a sequence is given by 2𝑛 + 2
Write the first five terms of the sequence. Work out the 100 -th term of the sequence. Explain whether 155 will occur in the sequence or not.
The 𝑛-th term of sequence is given as 4𝑛 − 2. Find the position for the term with the value 82.
A sequence is given as 3, 8, 13, 18, 23. Explain whether 387 is a term in the sequence or not.
The first five terms of an arithmetic sequence are 6, 11, 16, 21, 26. Find an expression in terms of
𝑛, for the 𝑛-th term of the sequence.
20
PRACTICE
Each table can fit a maximum of four chairs. Once the tables are pushed together the chairs where the
tables join can no longer be placed.
The image shows the layout of different numbers of tables with chairs.
Complete the table
Sara’s street party will need chairs for 115 people. Chairs cost ℒ2.00 each and tables cost ℒ10.00.
Work out how many tables Sara would need and use this to calculate the total cost.
Tables 1 2 3 4 Chairs 4 6 21 PRACTICE
The first five terms in a number sequence are 126, 122, 118, 114, 110.
Write down the next two terms in the sequence. If the 20 -th term of the sequence is 50. Write down the 21 -st term of the sequence.
A sequence has its first five terms as −1, 3, 7, 11, 15. The 𝑛-th term of another arithmetic
sequence is given as 8𝑛 − 16.
John says that there is a number that is in both the sequences. Is he right or wrong? Explain. 22
PRACTICE
A sequence has the 𝑛-th term as 𝑛ଶ^ + 6𝑛 − 10. List the first five terms of the sequence
The n-th term of a sequence is 𝑛ଶ^ − 2. A term in this sequence is 287. Find the position of this term
in the sequence.
Given the sequence 3, 6, 11, 18. What is the next term? Circle your answer
Circle the quadratic sequences from the list of sequences below
25 PRACTICE
Does the number 765 appear in the sequence defined by quadratic equation 𝑛ଶ^ + 6𝑛 − 10
Find the n-th term of the following sequence 100, 96, 90, 82, 72
A quadratic sequence is shown below
The sequence has an n-th term of 𝑛ଶ^ − 𝑛 + 5
Find the values of 𝑥 and 𝑦
26
PRACTICE
Each sequence below increase/decreases by the same amount each time. Find the missing terms
Terms 4 8 10 18 40 51 15 24 42 1 19 18 39 34 24 3 27 6 42 27 PRACTICE
Here are the first four terms of the sequence 9, 13, 17, 21. Work out the difference between 10th
and 15th^ term
The rule for continuing a sequence is multiply by 4 then subtract 5. The first term of the sequence is 3.
Find the next two terms of the sequence
A sequence is defined as 2, 6, 22, 86. The rule for continuing the sequence is multiply by 𝑎 then
subtract 𝑏
Find the values of 𝑎 and 𝑏 28
PRACTICE
The first term of a sequence is −5. The rule for continuing the sequence is first add 5 and then
multiply by 4. Find the next two terms of the sequence.
What is the 50 th term of the sequence 1.2𝑛 + 5
Is 53 in the sequence 3𝑛 + 2
The next two terms in the sequence
ଵ
ଶ , 2^
ଵ
ଶ , 12^
ଵ ଶ 31 PRACTICE
Lucy has a charm necklace with 30 charms on it. Every month she adds two new charms onto the
necklace. How many charms will there be on the necklace after two years?
Below is a number sequence found by counting the edges of the pattern.
What is the value of term 4. Write the n-th term of the sequence How many edges would be there in term 16 32
PRACTICE
A different geometric sequence is 2, 𝑎 , 288. Calculate the value of 𝑎
The n-th term of the geometric sequence
ଵ
ଷ ,^
ହ
,^
ଶହ
ଵଶ ,^
ଵଶହ
ଶସ ,^
ଶହ ସ଼
The next two terms in the sequence 0.02, 0.12, 0.
Josh says is a geometric progression 6, 12, 15, 30, 35, 70. Sofia says 4, 12, 36, 108, 324, 972 is a
geometric progression. Who is correct?
33 PRACTICE (HARD)
Bacteria divide by binary fission. Under optimum conditions, the bacteria Escherichia Coli divides
every 20 minutes. Sonia places 10 E. Coli bacteria into a petri dish and records the number of bacteria
she observes every 20 minutes for hours. Her results are shown below.
How many bacteria would Sonia expect to see after 4 hours?
Salmonella Enterica divides every 30 minutes. Sonia places 10 of these bacteria in a petri dish. How
long will it take for her to observe above 1,000,000 S. Enterica?
Salmonella symptoms start to appear 3 days after 1 bacteria enters the body. By this time, how many
bacteria could be produced?
Time 0 20 40 60 80 100 120 Bacteria 10 20 40 80 160 320 640 34
PRACTICE
Peter arranges blue and white tiles into patterns to make an arithmetic sequence. Below are the first 3
terms.
Write an expression for the number of blue tiles in Pattern 𝑛. Peter wants to create a pattern that has 31 white tiles. How many blue tiles would he need to complete the pattern? Luke says that it is possible to create a pattern using 82 blue tiles. Is Luke correct? Explain your answer. 37 PRACTICE
In a traditional mill, a water wheel drives a runner-stone to grind flour. For each turn of the water
wheel, the runner-stone revolves 2
ଵ
ସ times.
Generate the sequence to show the number of turns of the runner-stone for each turn of the water wheel. Write the n-th term of the sequence in the form × 𝑛 The water wheel takes 200 litres of water during each revolution. How much water is needed for the runner- stone to revolve 45 times? State the units in your answer. 38
PRACTICE
The first two terms of quadratic sequence are
10 and 17
Here is some information about the sequence
Work out an expression for the 𝑛-th term of
the sequence
39 PRACTICE
The first three terms of geometric progression are
ଶ
ଷ ,^
ସ
଼ (^) ଶ
Circle the fourth term
All terms of geometric progression are positive
The second and fourth terms are shown
Work out first and third terms Term-1 Term-2 Term-3 Term-
40
