Sequences

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课程笔记

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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练习题

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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

复习笔记

## 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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