Sequences

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Introduction to Sequences

A sequence is an ordered set of numbers that follow a rule (e.g., 3, 6, 9, 12… add 3 each time).
Each number in a sequence is called a term; its position is denoted by n (n=1 for first term).
A term-to-term rule tells how to get the next term from the current one (e.g., add 4, multiply by 3).
A position-to-term rule (nth term formula) lets you find any term directly: substitute n=1,2,3…
To check if a value belongs to a sequence, set the nth term equal to the value and solve for n; if n is a whole number, it is in the sequence.

Sequence position notation

nth Terms of Linear Sequences

A linear (arithmetic) sequence has a constant common difference (d) between consecutive terms.
The nth term formula is dn + b, where d is the common difference and b is the term before the first term (zero term).
To find b, continue the sequence backwards by one term (subtract d from the first term).
Example: For 5, 7, 9, 11… d=2, b=3 → nth term = 2n+3.
For decreasing sequences, d is negative (e.g., 15,10,5… d=-5, b=20 → nth term = -5n+20).

Quadratic Sequences

A quadratic sequence has an nth term involving n²; its second differences are constant.
To find the nth term: Step 1 – find first and second differences. Step 2 – halve the second difference to get a (coefficient of n²).
Step 3 – write out an² and subtract from the original sequence to get a linear sequence. Step 4 – find the nth term of that linear sequence (bn+c).
Step 5 – add an² + bn + c to get the nth term formula.
Example: For 6,9,14,21,30… second difference=2 → a=1; an²=1,4,9,16,25; difference=5,5,5,5,5 → linear nth term=5; so nth term = n²+5.
Common simple quadratics: n²+1, 2n², (n+1)²-2, etc.

Other Sequences

Cubic sequences have constant third differences; nth term involves n³. The coefficient a = (third difference)/6.
Exponential (geometric) sequences multiply by a constant common ratio (r) each time; nth term = a × rⁿ (or a × rⁿ⁻¹).
Common sequences: prime numbers (2,3,5,7…), triangular numbers (1,3,6,10…), cube numbers (1,8,27,64…).
Sequences can be combined: e.g., fractions with linear numerator and cubic denominator, or sums of two sequences.
Harder problems may involve setting up and solving equations (e.g., simultaneous equations) to find unknown terms or ratios.

Fibonacci and Special Sequences

In a Fibonacci sequence, each term is the sum of the two previous terms (e.g., 1,1,2,3,5,8…).
Given first two terms a and b, the 3rd term is a+b, 4th is a+2b, 5th is 2a+3b, 6th is 3a+5b.
To find unknown terms in a Fibonacci sequence, set up equations using given terms and solve.
Some sequences follow a pattern like adding a decreasing amount or using a specific rule (e.g., term-to-term rules).

Finding Terms and Checking Membership

To find a specific term, substitute n into the nth term formula (e.g., 20th term of n²+5 = 20²+5 = 405).
To check if a number is in a sequence, set the nth term equal to the number and solve for n; if n is a positive integer, it is a term.
Example: Is 98 in the sequence 8n+2? 8n+2=98 → n=12, yes. Is 124? 8n+2=124 → n=15.25, no.
Always write the position number above each term in the exam to help spot patterns.

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1.What is the next term in the sequence 8, 11, 14, 17, 20?
Easy
- A21
- B22
- C23
- D24
2.The nth term of a sequence is n² + 1. What are the first three terms?
Easy
- A2, 5, 10
- B1, 4, 9
- C2, 4, 8
- D1, 5, 10
3.Write an expression for the nth term of the sequence 15, 12, 9, 6.
Medium
- A18 - 3n
- B3n + 12
- C15 - 3n
- D3n + 15
4.A sequence has nth term 2n² + 5n - 15. Find the difference between the 4th term and the 5th term.
Medium
- A23
- B27
- C19
- D31
5.In a sequence, T1 = 17, T2 = 12, T3 = 7, T4 = 2. Find T5.
Easy
- A-3
- B-8
- C3
- D0
6.The nth term of a sequence is 60 - 8n. Find the largest number in this sequence.
Medium
- A52
- B60
- C56
- D48
7.Find an expression for the nth term of the sequence 12, 19, 26, 33, 40.
Medium
- A7n + 5
- B5n + 7
- C7n + 12
- D12n + 7
8.Find an expression for the nth term of the sequence 7, 5, 3, 1, -1, ...
Medium
- A9 - 2n
- B2n + 5
- C7 - 2n
- D2n + 7

Thêm câu hỏi

9.The first four terms of a sequence are 5, 8, 11, 14. Write down the next term and find an expression for the nth term.
Medium
- Anext 17, nth term 3n+2
- Bnext 16, nth term 3n+2
- Cnext 17, nth term 3n+1
- Dnext 16, nth term 3n+1
10.For the sequence 19, 15, 11, 7, ... write down the next two terms and find the nth term.
Medium
- Anext 3, -1; nth term 23-4n
- Bnext 4, 0; nth term 23-4n
- Cnext 3, -1; nth term 19-4n
- Dnext 4, 0; nth term 19-4n
11.Find the value of n when the nth term of the sequence 19, 15, 11, 7, ... is -65.
Medium
- A22
- B21
- C23
- D20
12.The sequence is 1/3, 3/4, 4/7, 7/11, 11/18, ... One term is p/q. Find the next term in terms of p and q.
Hard
- Aq/(p+q)
- B(p+q)/q
- Cp/(p+q)
- D(p+q)/p
13.Find the next term of the sequence 1/2, 3/4, 7/6, 13/8, 21/10, ...
Hard
- A31/12
- B31/10
- C29/12
- D33/12
14.Find the next term and the nth term of the sequence 3/5, 4/7, 5/9, 6/11, 7/13, ...
Hard
- Anext 8/15, nth term (n+2)/(2n+3)
- Bnext 8/15, nth term (n+2)/(2n+1)
- Cnext 9/17, nth term (n+2)/(2n+3)
- Dnext 8/15, nth term (n+1)/(2n+3)
15.The nth term of a sequence is given by an² + bn where a and b are integers. The 2nd term is -2 and the 4th term is 12. Find the 6th term.
Hard
- A42
- B38
- C46
- D34
16.Find an expression for the nth term of the quadratic sequence 0, 2, 6, 12, 20.
Medium
- An² - n
- Bn² + n
- Cn² - 1
- Dn² + 1
17.Find the next term in the sequence 7, 5, 3, 1, -1, ...
Easy
- A-3
- B-2
- C0
- D2
18.The nth term of a sequence is 3n + 2. What is the 5th term?
Easy
- A17
- B15
- C13
- D11
19.A sequence has nth term 4n - 3. Which of the following is the 1st term?
Easy
- A1
- B4
- C7
- D3
20.The first four terms of a sequence are 5, 8, 11, 14. What is the nth term?
Medium
- A3n + 2
- B2n + 3
- C3n + 5
- D5n
21.Find the next term in the sequence 1/2, 1/4, 1/6, 1/8, 1/10, ...
Easy
- A1/12
- B1/14
- C1/11
- D1/20
22.The nth term of a sequence is 4n² + n + 3. Find the 2nd term.
Medium
- A21
- B18
- C19
- D20
23.The sequence 9, 12, 17, 24, 33, ... has nth term of the form an² + bn + c. What is the next term?
Medium
- A44
- B42
- C48
- D40
24.The first three terms of a Fibonacci sequence are a, b, a+b. If the 3rd term is 7 and the 6th term is 29, find a and b.
Hard
- Aa=2, b=5
- Ba=3, b=4
- Ca=1, b=6
- Da=4, b=3
25.Find the nth term of the sequence 4, 8, 12, 16, 20, ...
Easy
- A4n
- Bn+4
- C4n+4
- Dn²
26.The sequence 1/3, 3/4, 4/7, 7/11, 11/18, ... has a term p/q. Find the next term in terms of p and q.
Hard
- A(p+q)/(p+2q)
- B(p+q)/q
- Cq/(p+q)
- D(p+q)/(2p+q)
27.A sequence has nth term 3n + 2. Which term of the sequence is 47?
Easy
- A15th
- B14th
- C16th
- D13th
28.The first four terms of a quadratic sequence are 4, 9, 16, 25. What is the nth term?
Medium
- A(n+1)²
- Bn² + 3
- Cn² + 2n + 1
- Dn² + 1
29.A sequence has first term 5 and term-to-term rule 'add 7'. What is the 10th term?
Easy
- A68
- B75
- C61
- D72
30.The nth term of a sequence is 2n² - 5. Which term is 67?
Medium
- A6th
- B7th
- C5th
- D8th
31.What is the next term in the sequence 2, 6, 18, 54, ...?
Easy
- A162
- B108
- C216
- D270
32.The nth term of a sequence is 5n - 3. Which of the following numbers is in the sequence?
Easy
- A40
- B37
- C44
- D33
33.The first three terms of a Fibonacci sequence are a, b, a+b. If the 4th term is 13 and the 5th term is 21, find a.
Medium
- A5
- B8
- C3
- D2
34.A sequence has nth term n² + n + 1. Find the difference between the 5th and 3rd terms.
Medium
- A20
- B18
- C22
- D16
35.The sequence 1/2, 1/4, 1/6, 1/8, 1/10, ... has nth term:
Easy
- A1/(2n)
- B1/(n+1)
- C2/n
- D1/(2n+2)
36.What is the next term in the sequence 22, 17, 12, 7, 2, ...?
Easy
- A-3
- B0
- C-5
- D3
37.The sequence 3, 6, 9, 12, ... has nth term 3n.
Easy
True or false?
38.A sequence has nth term n² + 5. What is the 4th term?
Easy
- A21
- B16
- C20
- D25
39.Which of the following are linear sequences? (Select all that apply)
Medium
- A2, 5, 8, 11, 14
- B1, 4, 9, 16, 25
- C10, 7, 4, 1, -2
- D3, 6, 12, 24, 48
40.Arrange the following steps in order to find the nth term of a quadratic sequence.
Medium
- Find the first and second differences.
- Divide the second difference by 2 to find the coefficient of n².
- Subtract an² from the original sequence.
- Find the nth term of the resulting linear sequence.
- Add the linear nth term to an² to get the quadratic nth term.
41.The first four terms of a quadratic sequence are 5, 7, 11, 17. What is the nth term?
Medium
- An² + 4
- Bn² - n + 5
- Cn² + n + 3
- D2n² + 3
42.Match each sequence to its type.
Medium
- 5, 10, 20, 40, ...
- 1, 4, 9, 16, ...
- 2, 5, 8, 11, ...
- 1, 8, 27, 64, ...
- Linear
- Quadratic
- Cubic
- Exponential
43.Arrange the steps to find the nth term of the quadratic sequence 5, 7, 11, 17, 25, ...
Medium
- Find the first and second differences.
- Halve the second difference to get a.
- Write down the sequence an² and subtract from original.
- Find the nth term of the linear differences.
- Add an² to the linear nth term.
44.Which of the following are linear sequences? (Select all that apply).
Easy
- A3, 6, 9, 12, ...
- B2, 4, 8, 16, ...
- C10, 7, 4, 1, ...
- D1, 4, 9, 16, ...
45.The sequence 1, 4, 9, 16, 25, ... has nth term n².
Easy
True or false?
46.Find the 80th term of the sequence with nth term 3n + 2.
Easy
47.Complete the sentence.
Easy
In a quadratic sequence, the ____ differences are constant.
48.What is the name for the amount each term increases or decreases by in a linear sequence?
Easy
49.Find the 5th term of the cubic sequence with nth term 2n³ + 3.
Easy
50.Find the next term in the sequence 3, 8, 13, 18, ...
Easy
51.The first three terms of an exponential sequence are x-1, 2x, x². By forming and solving an equation, find the common ratio r, given that x ≠ 0.
Hard
52.A sequence has nth term an² + bn + c. The first three terms are 2, 6, 12. Find the values of a, b, and c.
Hard
53.Find the 80th term of the sequence with nth term 3n + 2.
Easy
- A242
- B290
- C266
- D218
54.Find the 5th term of the cubic sequence with nth term 2n³ + 3.
Easy
- A303
- B278
- C253
- D228
55.Find the next term in the sequence 3, 8, 13, 18, ...
Easy
- A23
- B25
- C21
- D27

Ghi chú ôn tập

## Sequences - Night Before Exam Summary ### Key Definitions - **Sequence**: ordered set of numbers following a rule. - **Term**: each number in a sequence. - **Position**: location of a term, denoted by n (n=1 is 1st term). - **nth term formula**: algebraic expression in n to find any term directly. ### Linear Sequences (Arithmetic) - Constant difference d between terms. - nth term = dn + b, where d = common difference, b = term before 1st term (zero term). - Example: 5,7,9,11,... → d=2, b=3 → nth term = 2n+3. ### Quadratic Sequences - Second differences are constant. - nth term = an² + bn + c. - Steps: 1. Find first and second differences. 2. a = (second difference) ÷ 2. 3. Write an² terms, subtract from original sequence. 4. Find nth term of the linear difference (bn + c). 5. Add: nth term = an² + bn + c. - Example: 6,9,14,21,30,... → second diff=2 → a=1 → an²=1,4,9,16,... → differences 5,5,5,... → nth term = n²+5. ### Cubic Sequences - Third differences constant. - nth term = an³ + bn² + cn + d, with a = (third difference)/6. - Compare to cube numbers n³. ### Exponential (Geometric) Sequences - Multiply by common ratio r each time. - nth term = a × rⁿ (or a × rⁿ⁻¹). - Example: 3,6,12,24,... → r=2 → nth term = 3×2ⁿ⁻¹. ### Other Sequences - **Fibonacci**: next term = sum of two previous terms (e.g., 1,1,2,3,5,8,...). - **Fractions**: treat numerator and denominator as separate sequences. - **Checking if a number is in a sequence**: set nth term formula equal to number, solve for n; n must be a positive integer. ### Tips - Write position numbers above terms. - Memorise square numbers 1² to 15². - For quadratic, always halve second difference to get a. - For exponential, divide any term by previous term to find r.

Sequence: An ordered set of numbers that follow a rule.
Term: Each number in a sequence.
Position: The location of a term within a sequence, denoted by n.
nth term formula: An algebraic expression in n that allows you to find any term in the sequence directly.
Linear sequence: A sequence where the difference between consecutive terms is constant; also called arithmetic sequence.
Common difference: The constant amount added (or subtracted) to get from one term to the next in a linear sequence.
Quadratic sequence: A sequence where the second differences are constant; its nth term formula involves n².
Cubic sequence: A sequence where the third differences are constant; its nth term formula involves n³.
Exponential sequence: A sequence where each term is multiplied by a constant (common ratio) to get the next term; also called geometric sequence.
Common ratio: The constant multiplier in an exponential sequence.
Fibonacci sequence: A sequence where each term is the sum of the two previous terms.

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