Sequences

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Notes de leçon

Terms and rules

A sequence is a list of numbers that follows a rule. Each number is a term; u_n or T(n) is the nth term.
A term-to-term rule says how to get from one term to the next (e.g. add 4). A position-to-term rule (the nth term) gives any term directly.
Here positions are positive integers starting at n = 1.

Arithmetic sequences

An arithmetic (linear) sequence goes up or down by a constant common difference d.
The nth term is dn + (a - d), where a is the first term. For 5, 8, 11, 14: d = 3, so 3n + 2.
To check whether a number is in the sequence, solve 3n + 2 = \text{number} and see if n is a positive integer.

Geometric sequences

A geometric sequence multiplies by a constant common ratio r: 3, 6, 12, 24 has r = 2. The nth term is ar^n-1.
Fibonacci-type sequences add the previous two terms.

Quadratic sequences

A constant nonzero second difference identifies a quadratic sequence an² + bn + c, with a equal to half that difference. Zero second difference gives a linear or constant sequence.
For 2, 7, 14, 23: first differences 5, 7, 9; second difference 2, so a = 1. Subtract n² (1, 4, 9, 16) to leave 1, 3, 5, 7 = 2n - 1. So n² + 2n - 1.

Differences in a quadratic sequence

Special sequences

Square numbers 1, 4, 9, 16; cube numbers 1, 8, 27, 64; triangular numbers 1, 3, 6, 10; powers of 2: 2, 4, 8, 16.

Diapos

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Questions d'entraînement

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1.What do we call each number in a sequence?
Easy
- Aa term
- Ba factor
- Ca power
- Da root
2.What is the next term of 2, 5, 8, 11?
Easy
- A13
- B14
- C15
- D12
3.What is the common difference of 10, 7, 4, 1?
Easy
- A3
- B-3
- C7
- D-7
4.What is the next term of 3, 6, 12, 24?
Easy
- A30
- B36
- C48
- D72
5.What is the nth term of 5, 8, 11, 14?
Easy
- A3n + 2
- B3n + 5
- Cn + 3
- D5n
6.What is the nth term of 10, 20, 30, 40?
Easy
- An + 10
- B10n
- C10n + 10
- Dn¹⁰
7.What is the 10th term of 2n + 5?
Easy
- A20
- B25
- C15
- D30
8.What is the nth term of 20, 17, 14, 11?
Easy
- A23 - 3n
- B3n + 17
- C20 - 3n
- D3n - 23

Plus de questions

9.Which is the 5th term of n² + 1?
Easy
- A11
- B26
- C25
- D10
10.Which is a square number?
Easy
- A15
- B25
- C35
- D45
11.Which is a cube number?
Easy
- A9
- B27
- C36
- D50
12.What is the next triangular number after 1, 3, 6, 10?
Easy
- A12
- B15
- C14
- D16
13.Is 100 a term of 3n - 2?
Medium
- Ayes, n = 34
- Byes, n = 33
- Cno
- Dyes, n = 35
14.Is 50 a term of 4n + 1?
Medium
- Ayes
- Bno, n = 12.25
- Cno, n = 13
- Dyes, n = 12
15.The 3rd term of a geometric sequence is 12 and the ratio is 2. What is the first term?
Medium
- A3
- B6
- C4
- D24
16.What is the nth term of 2, 6, 18, 54?
Medium
- A2 \times 3^n-1
- B2n
- C3 \times 2ⁿ
- D2 + 4n
17.What are the next two terms of 1, 1, 2, 3, 5?
Medium
- A7, 9
- B8, 13
- C6, 9
- D8, 11
18.What is the second difference of 1, 4, 9, 16, 25?
Medium
- A1
- B2
- C3
- D4
19.Find the nth term of 2, 7, 14, 23.
Medium
- An² + 2n - 1
- B2n²
- Cn² + 5
- D3n - 1
20.Find the nth term of 5, 12, 23, 38.
Medium
- A2n² + n + 2
- Bn² + 4
- C3n² + 2
- D2n² + 3
21.The nth term of a sequence is n² - 3n. What is the 6th term?
Medium
- A12
- B18
- C6
- D36
22.Sequences A: 2, 4, 6, 8 and B: 1, 4, 9, 16. Which is quadratic?
Medium
- AA
- BB
- Cboth
- Dneither
23.The nth term is 5n - 3. What is the first term?
Easy
- A2
- B5
- C-3
- D8
24.A matchstick pattern uses 4, 7, 10 sticks for patterns 1, 2, 3 and adds 3 sticks each time. How many sticks does pattern 5 use?
Easy
- A16
- B15
- C19
- D13
25.For the sequence 4, 7, 10, 13, which expression gives the 20th term?
Medium
- A3 \times 20 + 1
- B3 \times 20
- C4 \times 20
- D20²
26.Which sequence is geometric?
Medium
- A2, 4, 6, 8
- B2, 4, 8, 16
- C1, 4, 9, 16
- D1, 2, 4, 7
27.In an arithmetic sequence, the difference between terms is constant.
Easy
True or false?
28.The sequence 1, 4, 9, 16 has constant first differences.
Easy
True or false?
29.The nth term of 3, 5, 7, 9 is 2n + 1.
Easy
True or false?
30.A geometric sequence adds the same amount each time.
Easy
True or false?
31.A number is a term of a linear sequence if solving the nth term equal to it gives a positive integer.
Medium
True or false?
32.Which are arithmetic sequences?
Medium
- A1, 3, 5, 7
- B2, 4, 8, 16
- C10, 7, 4, 1
- D1, 4, 9, 16
- E5, 5, 5, 5
33.Which are square numbers?
Medium
- A16
- B18
- C25
- D36
- E40
34.Find the nth term of the sequence 7, 11, 15, 19 and use it to find the 50th term.
Medium

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