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... rule: add 3).
Each number is a term; its position is given by n (n=1 is 1st term).
A term-to-term rule tells how to get the next term (e.g., add 4 each time).
A position-to-term rule (nth term formula) lets you find any term directly (e.g., 8n+2).
To check if a value belongs to a sequence, set the nth term formula equal to the value and solve for n; if n is a whole number, it is a term.

Sequence position notation

Continuing Sequences

Use first differences to spot patterns: e.g., 4,7,10,13 → differences +3, next term 16.
If first differences change by a constant amount (second differences constant), the sequence is quadratic.
If first differences double each time, the next difference is double the last.
Be familiar with special sequences: square numbers (1,4,9,16...), cube numbers (1,8,27,64...), triangular numbers (1,3,6,10...).

Triangular numbers as dots

nth Terms of Linear Sequences

A linear sequence (arithmetic) has a constant common difference d.
The nth term formula is dn + b, where d is the common difference and b is the term before the 1st term (zero term).
To find b, continue the sequence backwards by one term (subtract d from the 1st term).
Example: 5,7,9,11 → d=2, b=3 → nth term = 2n+3.

Quadratic Sequences

A quadratic sequence has an nth term involving n²; its second differences are constant.
Compare to square numbers (n²): e.g., 6,9,14,21,30 → each term is 5 more than n² → nth term = n²+5.
For simple quadratics an²+b, a = ½ × (second difference).
Example: sequence 3,9,19,33,51 → second difference = 4 → a=2, then find b by comparing to 2n².

Cubic Sequences

A cubic sequence has an nth term involving n³; its third differences are constant.
Compare to cube numbers (n³): e.g., 2,9,28,65,126 → each term is 1 more than n³ → nth term = n³+1.
For simple cubics an³+b, a = ⅙ × (third difference).
Know cube numbers up to 5³=125 and 10³=1000.

Finding the nth Term: Worked Examples

Linear: Find nth term of -7,-3,1,5,9 → d=4, b=-11 → nth term = 4n-11.
Quadratic: Find nth term of 6,9,14,21,30 → second diff=2 → a=1 → compare to n² → nth term = n²+5.
Cubic: Sequence 4,25,82,193,376 → third diff=18 → a=3 → compare to 3n³ → nth term = 3n³+1.
Always check your formula by substituting n=1,2,3 to see if it matches the given terms.

Using nth Term to Find Terms and Check Membership

To find a specific term, substitute n into the formula (e.g., 20th term of n²+5 = 400+5=405).
To check if a number is in the sequence, set formula = number and solve for n; if n is a positive integer, it is a term.
Example: Is 98 in sequence 8n+2? 8n+2=98 → n=12 → yes, 12th term.
If n is not an integer (e.g., n=15.25), the number is not in the sequence.

Pattern Sequences (Exam Style)

Many exam questions involve patterns of counters, lines, or squares; find the nth term for each quantity.
Example: Pattern n has black counters = 2n+2, white counters = n².
Use the nth term to check if a pattern can be made with given resources (e.g., 30 black, 140 white for Pattern 12).
For linear patterns, the number of rows or houses can be found using the nth term formula.

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練習問題

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1.What is the next term in the sequence 18, 21, 26, 33, 42, ...?
Easy
- A53
- B51
- C55
- D49
2.Find the next term in the sequence 18, 20, 24, 32, 48, ...
Medium
- A80
- B64
- C56
- D72
3.The first four terms of a sequence are 17, 10, 3, -4. What is the next term?
Easy
- A-11
- B-10
- C-12
- D-9
4.The nth term of a sequence is n³ - 5. What is the first term?
Easy
- A-4
- B-3
- C0
- D1
5.The first four terms of a sequence are 29, 32, 35, 38. What is the next term?
Easy
- A41
- B40
- C42
- D39
6.The first four terms of a sequence are 32, 27, 22, 17. What is the next term?
Easy
- A12
- B13
- C11
- D10
7.The nth term of a sequence is n² + 5. What is the second term?
Easy
- A9
- B6
- C7
- D10
8.Find the next term in the sequence 12, 7, 2, -3, -8, ...
Easy
- A-13
- B-12
- C-14
- D-11

もっと問題

9.The first four terms of a sequence are 8, 15, 22, 29. What is an expression for the nth term?
Medium
- A7n + 1
- B7n - 1
- C8n
- D7n + 8
10.Find the first three terms of the sequence with nth term n² + 5n.
Medium
- A6, 14, 24
- B6, 12, 20
- C5, 14, 24
- D6, 14, 26
11.The first four terms of a sequence are -2, 2, 6, 10. Find an expression for the nth term.
Medium
- A4n - 6
- B4n - 2
- C4n + 2
- D4n - 10
12.The nth term of a sequence is 60 - 8n. What is the largest number in this sequence?
Medium
- A52
- B60
- C56
- D44
13.The first five terms of a sequence are 12, 19, 26, 33, 40. Find an expression for the nth term.
Medium
- A7n + 5
- B7n - 5
- C7n + 12
- D7n + 2
14.The nth term of a sequence is n² + 5. Is 261 a term in this sequence?
Hard
- AYes, it is the 16th term
- BYes, it is the 15th term
- CNo
- DYes, it is the 17th term
15.The first four terms of a sequence are 27, 33, 39, 45. Find the nth term.
Medium
- A6n + 21
- B6n + 27
- C6n + 15
- D6n + 9
16.The first three terms of a sequence are 3, 9, 15, ... Find the nth term.
Medium
- A6n - 3
- B6n + 3
- C3n
- D6n - 9
17.Find the next term in the sequence: 4, 7, 10, 13, ...
Easy
- A16
- B17
- C15
- D14
18.Find the next term in the sequence: 6, 1, -4, -9, ...
Easy
- A-14
- B-13
- C-15
- D-12
19.Find the next term in the sequence: 2, 8, 15, 23, ...
Medium
- A32
- B31
- C30
- D33
20.Find the next term in the sequence: 1, 3, 7, 15, ...
Medium
- A31
- B29
- C30
- D32
21.The nth term of a sequence is 3n + 2. What is the first term?
Easy
- A5
- B3
- C2
- D8
22.The nth term of a sequence is 2n - 3. What is the second term?
Easy
- A1
- B2
- C-1
- D4
23.The first four terms of a sequence are 5, 8, 11, 14. Find the nth term.
Medium
- A3n + 2
- B3n + 5
- C2n + 3
- D3n - 2
24.Find the next term in the sequence: 3, 6, 11, 18, 27, ...
Medium
- A38
- B37
- C36
- D39
25.A theatre has three sections of seats: A, B and C. Section A has 152 seats. In Section A, there are 12 seats in the front row and each row has 2 more seats than the row in front of it. How many rows are there in Section A?
Medium
- A8 rows
- B9 rows
- C10 rows
- D11 rows
26.The first four terms of a sequence are 3, 11, 19, 27. Find an expression for the nth term.
Easy
- A8n - 5
- B8n + 3
- C8n - 3
- D8n + 5
27.A road has houses numbered from 1 to 349. Houses on the West side have odd numbers. Tomaz delivers to every house on the West side starting at house 1. Find an expression for the house number of the nth house he delivers to.
Medium
- A2n - 1
- B2n + 1
- Cn + 1
- D2n
28.A sequence is defined by the nth term = n² + 2n. Find the first three terms.
Easy
- A3, 8, 15
- B3, 6, 11
- C1, 4, 9
- D3, 8, 13
29.Find the next term in the sequence: 4, 7, 13, 25, 49, ...
Medium
- A97
- B73
- C85
- D101
30.The nth term of a sequence is 60 - 8n. Find the largest number in this sequence.
Medium
- A52
- B60
- C56
- D44
31.Here are the first five terms of a sequence: 3, 11, 19, 27, 35. Show that 451 is a term in the sequence.
Medium
- AYes, it is the 57th term
- BYes, it is the 56th term
- CNo, it is not a term
- DYes, it is the 55th term
32.A sequence of patterns is made using black and white counters. The number of black counters in Pattern n is 2n + 2. The number of white counters in Pattern n is n². Can Elena make Pattern 12 using 30 black counters and 140 white counters?
Hard
- AYes, she has enough counters
- BNo, she needs more black counters
- CNo, she needs more white counters
- DNo, she needs more of both
33.A sequence has nth term = n² + 2n. Find the 20th term.
Medium
- A440
- B420
- C400
- D460
34.The first three terms of a sequence are 2, 9, 28. Find the nth term.
Medium
- An³ + 1
- Bn³ - 1
- C3n - 1
- Dn² + 1
35.What are the first three terms of the sequence with nth term n³ - 5?
Easy
- A-4, 3, 22
- B1, 8, 27
- C-5, -4, -1
36.Find the next term in the sequence: 16, 19, 25, 37, 61, ...
Medium
- A109
- B85
- C97
- D91
37.A sequence has nth term = 3n + 2. What is the 10th term?
Easy
- A32
- B30
- C35
- D28
38.What is the next term in the sequence 22, 17, 12, 7, 2, ...?
Easy
- A-3
- B0
- C3
- D-2
39.The sequence 3, 6, 11, 18, 27, ... is a quadratic sequence.
Medium
True or false?
40.Which of the following are sequences with a common difference? (Select all that apply)
Medium
- A5, 8, 11, 14, ...
- B2, 6, 18, 54, ...
- C10, 7, 4, 1, ...
- D1, 4, 9, 16, ...
- E20, 15, 10, 5, ...
41.Match each sequence to its nth term formula.
Medium
- 2, 5, 8, 11, ...
- 3, 6, 12, 24, ...
- 1, 4, 9, 16, ...
- 3n - 1
- n²
- 3 × 2^n-1
42.What type of sequence has constant second differences?
Easy
- ALinear
- BQuadratic
- CCubic
- DGeometric
43.Put the following steps in order to find the nth term of a linear sequence.
Medium
- Find the common difference d.
- Write the formula as dn + b.
- Work backwards to find the term before the first term, b.
- Substitute n=1 to check.
44.A quadratic sequence has constant second differences.
Easy
True or false?
45.What are the next two terms in the sequence 5, 12, 19, 26, ...?
Easy
- A33, 40
- B32, 39
- C33, 41
- D34, 41
46.Complete the sentence.
Easy
In a linear sequence, the difference between consecutive terms is called the ____ difference.
47.The nth term of a sequence is n² + 5. What is the 10th term?
Easy
48.The first four terms of a sequence are 2, 9, 28, 65. Find the nth term formula, and hence find the 5th term.
Medium
49.A sequence has nth term = n² + 2n. Show that 195 is a term in this sequence.
Hard
50.Find the 10th term of the sequence with nth term 5n - 3.
Easy
51.A sequence has nth term = n³ - 5. Determine whether 221 is a term in this sequence.
Medium
52.Here are the first five terms of a sequence: 3, 8, 15, 24, 35. Find an expression for the nth term.
Medium
53.The nth term of a sequence is n² + 5. What is the 10th term?
Easy
- A105
- B127
- C116
- D94
54.The first four terms of a sequence are 2, 9, 28, 65. Find the nth term formula, and hence find the 5th term.
Medium
- A152
- B126
- C113
- D139
55.Find the 10th term of the sequence with nth term 5n - 3.
Easy
- A47
- B52
- C42
- D57

復習ノート

## Sequences - Quick Revision ### Key Definitions - **Sequence**: ordered set of numbers following a rule. - **Term**: each number in a sequence. - **Position**: location of a term (n = 1 for 1st term). - **nth term**: formula to find any term directly. ### Term-to-Term Rule Tells how to get the next term from the current one (e.g., add 4 each time). ### Position-to-Term Rule (nth term) Substitute n = 1, 2, 3... into the formula. ### Linear Sequences - Common difference d (same amount added/subtracted each time). - nth term = dn + b, where b is the term before the 1st term (zero term). - Example: 5,7,9,11 → d=2, b=3 → nth term = 2n+3. ### Quadratic Sequences - Second differences constant. - Compare to square numbers (n²). - Simple form: nth term = n² + c or an² + b. - a = half of second difference. ### Cubic Sequences - Third differences constant. - Compare to cube numbers (n³). - Simple form: nth term = n³ + c or an³ + b. - a = 1/6 of third difference. ### Checking if a Number is in a Sequence Set nth term formula equal to the number; solve for n. If n is a positive integer, it's a term. ### Common Sequences to Know - Squares: 1,4,9,16,25,36,49,64,81,100,121,144,... - Cubes: 1,8,27,64,125,... - Triangular: 1,3,6,10,15,... ### Worked Example (Linear) Find nth term of -7,-3,1,5,9,... d = 4, b = -11 → nth term = 4n - 11. ### Worked Example (Quadratic) Sequence: 6,9,14,21,30,... First differences: 3,5,7,9,... Second differences: 2,2,2,... Compare to n²: 1,4,9,16,25 → each term is 5 more → nth term = n² + 5.

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 (n=1 for 1st term).
nth term: A formula (in terms of n) that gives the value of the term at position n.
Term-to-term rule: A rule that tells how to get the next term from the current term (e.g., add 3 each time).
Common difference: The constant amount added (or subtracted) to get from one term to the next in a linear sequence.
Linear sequence: A sequence where the difference between consecutive terms is constant (also called arithmetic sequence).
Quadratic sequence: A sequence where the second differences are constant; its nth term involves n².
Cubic sequence: A sequence where the third differences are constant; its nth term involves n³.

フラッシュカード

What is a sequence?

A sequence is an ordered set of numbers that follow a rule.

What is the term-to-term rule for the sequence 4, 14, 24, 34, ...?

Add 10 each time.

What is the nth term formula for a linear sequence?

nth term = dn + b, where d is the common difference and b is the term before the first term.

Find the next term in the sequence: 18, 21, 26, 33, 42, ...

53 (differences increase by 2 each time: +3, +5, +7, +9, next +11).

Find the next term in the sequence: 18, 20, 24, 32, 48, ...

80 (differences double: +2, +4, +8, +16, next +32).

The first four terms of a sequence are 17, 10, 3, -4. Find the next term and the term-to-term rule.

Next term: -11. Rule: subtract 7 each time.

The nth term of a sequence is n³ − 5. Write down the first three terms.

1³−5 = -4, 2³−5 = 3, 3³−5 = 22. So -4, 3, 22.

The nth term of a sequence is n² + 5. Find the first three terms.

1²+5=6, 2²+5=9, 3²+5=14. So 6, 9, 14.

Find the next term in the sequence: 12, 7, 2, -3, -8, ...

-13 (subtract 5 each time).

Find the next term in the sequence: 4, 7, 13, 25, 49, ...

97 (differences double: +3, +6, +12, +24, next +48).

The first four terms of a sequence are 8, 15, 22, 29. Find the nth term.

nth term = 7n + 1 (common difference 7, term before first is 1).

Find the first three terms of the sequence with nth term n² + 5n.

1²+5(1)=6, 2²+5(2)=14, 3²+5(3)=24. So 6, 14, 24.

The first four terms of a sequence are -2, 2, 6, 10. Find the nth term.

nth term = 4n - 6 (common difference 4, term before first is -6).

The nth term of a sequence is 60 - 8n. Find the largest number in this sequence.

Substitute n=1: 60-8=52. The sequence decreases, so largest is 52.

Here are the first five terms of a sequence: 12, 19, 26, 33, 40. Find the nth term.

nth term = 7n + 5 (common difference 7, term before first is 5).

Show that 261 is a term in the sequence with nth term n² + 5.

Set n²+5=261 → n²=256 → n=16 (whole number). So 261 is the 16th term.

The first four terms of a sequence are 27, 33, 39, 45. Find the nth term.

nth term = 6n + 21 (common difference 6, term before first is 21).

Find the first three terms of the sequence with nth term n² + 2n.

1²+2(1)=3, 2²+2(2)=8, 3²+2(3)=15. So 3, 8, 15.

The first four terms of a sequence are 3, 9, 15, 21. Find the nth term.

nth term = 6n - 3 (common difference 6, term before first is -3).

The first four terms of a sequence are 5, 8, 11, 14. Find the nth term.

nth term = 3n + 2 (common difference 3, term before first is 2).

The first five terms of a sequence are 3, 11, 19, 27, 35. Find the nth term and show that 451 is a term.

nth term = 8n - 5. Set 8n-5=451 → 8n=456 → n=57 (whole number). So 451 is the 57th term.

What is a quadratic sequence?

A quadratic sequence has an nth term formula involving n², and the second differences are constant.

Find the nth term of the sequence 6, 9, 14, 21, 30, ...

nth term = n² + 5 (second difference is 2, so n²; compare to squares: each term is 5 more).

What is a cubic sequence?

A cubic sequence has an nth term formula involving n³, and the third differences are constant.

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