Sequences and the nth term

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课程笔记

What is a sequence?

A sequence is an ordered list of numbers, called terms.
Each term has a position: 1st, 2nd, 3rd, etc. We can write the term in position n as a_n.
A term-to-term rule tells you how to get from one term to the next, e.g. 'add 3' or 'multiply by 2'.
A position-to-term rule (or nth term) lets you find any term directly from its position number n, without working through the list.

Arithmetic (linear) sequences

An arithmetic sequence (also called a linear sequence) goes up or down by the same amount each time.
The constant difference between consecutive terms is the common difference, often called d.
Example: 5, 7, 9, 11, 13, ... has common difference d = 2.
To check if a sequence is arithmetic, subtract each term from the next; if the answer is always the same, it is arithmetic.

Finding the nth term of an arithmetic sequence

For an arithmetic sequence with first term a₁ and common difference d, the nth term is: a_n = a₁ + (n − 1)d.
Here a₁ is the first term, d is the common difference, and n is the position number.
Example: for 5, 7, 9, 11, ... a₁ = 5 and d = 2, so a_n = 5 + (n − 1) × 2 = 2n + 3.
You can use the nth term to find any term: for n = 10, a₁₀ = 2 × 10 + 3 = 23.

Deciding if a number is in a sequence

To test whether a number belongs to an arithmetic sequence, set the nth term equal to that number and solve for n.
If n comes out as a positive whole number, the number is in the sequence; otherwise it is not.
Example: is 100 in the sequence with nth term 2n + 3? Solve 2n + 3 = 100 → 2n = 97 → n = 48.5, so 100 is not in the sequence.
Example: is 101 in the same sequence? 2n + 3 = 101 → 2n = 98 → n = 49, so 101 is the 49th term.

Geometric sequences

A geometric sequence is made by multiplying by the same number each time.
The multiplier is called the common ratio.
Example: 3, 6, 12, 24, 48, ... has common ratio 2.
To find the next term, multiply the previous term by the common ratio.

Special sequences

Square numbers: 1, 4, 9, 16, 25, ... (the nth term is n²).
Cube numbers: 1, 8, 27, 64, 125, ... (the nth term is n³).
Triangular numbers: 1, 3, 6, 10, 15, ... (the nth term is n(n + 1)/2).
Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ... where each term is the sum of the two previous terms.

Simple quadratic sequences

In a quadratic sequence, the differences between terms change, but the second differences are constant.
The nth term of a quadratic sequence contains an n² term, for example a_n = n² + 2.
Example: 3, 6, 11, 18, 27, ... has first differences 3, 5, 7, 9 (increasing by 2 each time), so it is quadratic.
To find the nth term, look for the n² pattern and then adjust by adding or subtracting a constant or linear term.

Summing an arithmetic series

The sum of a finite arithmetic sequence is called an arithmetic series.
The sum of the first n terms is: S_n = n(a₁ + a_n) / 2, where a₁ is the first term and a_n is the nth term.
Example: 2 + 5 + 8 + 11 + 14 has n = 5, a₁ = 2, a₅ = 14, so S₅ = 5(2 + 14)/2 = 40.
This formula works for any arithmetic progression of real numbers.

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练习题

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1.In an arithmetic sequence, what name is given to the constant difference between successive terms?
Easy
- Acommon difference
- Bcommon ratio
- Cnth term
- Dposition-to-term rule
2.The first term of an arithmetic sequence is 5 and the common difference is 7. Which expression gives the nth term?
Easy
- A5 + 7n
- B5 + 7(n - 1)
- C7 + 5(n - 1)
- D5n + 7
3.The sequence 2, 4, 6, 8, ... is an arithmetic sequence with common difference 2.
Easy
True or false?
4.Which of these sequences are arithmetic? (select all that apply)
Medium
- A3, 7, 11, 15, ...
- B1, 2, 4, 8, ...
- C10, 7, 4, 1, ...
- D1, 4, 9, 16, ...
- E5, 5, 5, 5, ...
5.The first term of an arithmetic sequence is 2 and the 10th term is 29. What is the common difference?
Medium
- A2.7
- B3
- C2
- D5
6.Which of these numbers is a term of the sequence with nth term 4n - 3?
Medium
- A15
- B18
- C21
- D23
7.Put these steps in the correct order to find the nth term of the arithmetic sequence 7, 11, 15, 19, ...
Medium
- Write the nth term as 4n + c.
- Find the common difference: 11 - 7 = 4.
- Substitute n = 1: 4(1) + c = 7, so c = 3.
- The nth term is 4n + 3.
8.Match each sequence to its nth term.
Medium
- 2, 5, 8, 11, ...
- 3, 6, 9, 12, ...
- 1, 3, 5, 7, ...
- 4, 7, 10, 13, ...
- 3n
- 2n - 1
- 3n + 1

复习笔记

## Sequences and the nth term A **sequence** is an ordered list of numbers. Each number is a **term**. You can build a sequence with a **term-to-term rule** (what you do to get the next term) or a **position-to-term rule** (a formula for the term in position n). ### Arithmetic (linear) sequences An **arithmetic sequence** adds or subtracts the same amount each time. That amount is the **common difference**, d. Example: 5, 7, 9, 11, 13, 15, ... has d = 2. If the first term is a_1 and the common difference is d, the nth term is: \(a_n = a_1 + (n-1)d\) (a_n = nth term, a_1 = first term, d = common difference, n = position number). **Finding the nth term:** 1. Find d (the jump between terms). 2. The formula starts with dn. 3. Adjust by adding or subtracting to make the first term correct. Example: 5, 7, 9, 11, ... has d = 2, so try 2n. For n = 1, 2n = 2, but the first term is 5, so add 3. nth term = 2n + 3. **Is a number in the sequence?** Put the number equal to the nth-term formula, solve for n. If n is a whole number, yes; if not, no. ### Other sequences - **Geometric sequences** multiply by the same number each time (for example 3, 6, 12, 24, ... multiply by 2). - **Square numbers:** 1, 4, 9, 16, 25, ... (n^2) - **Cube numbers:** 1, 8, 27, 64, 125, ... (n^3) - **Triangular numbers:** 1, 3, 6, 10, 15, ... - **Fibonacci numbers:** 1, 1, 2, 3, 5, 8, ... (each term is the sum of the two before it) - **Simple quadratic sequences** have a constant second difference; the nth term contains n^2. ### Sum of an arithmetic series A **finite arithmetic progression** is a finite portion of an arithmetic sequence. Its sum is an **arithmetic series**. For n terms from a_1 to a_n: \(S_n = \frac{n(a_1 + a_n)}{2}\) (S_n = sum of n terms, a_1 = first term, a_n = nth term, n = number of terms). Example: 2 + 5 + 8 + 11 + 14 = 40, because \(\frac{5(2+14)}{2} = \frac{5 \times 16}{2} = 40\). This formula works for any arithmetic progression of real numbers. ### Exam tips - Always check the first term after writing a formula. - For linear sequences, the coefficient of n is the common difference. - For quadratic sequences, look at the second difference. - To test whether a number belongs, solve for n and check it is a positive whole number.

Sequence: An ordered list of numbers, where each number is called a term.
Term: A single number in a sequence.
Term-to-term rule: A rule that tells you how to get from one term to the next.
Position-to-term rule: A rule or formula that gives the term from its position number n.
Arithmetic (linear) sequence: A sequence where the difference between consecutive terms is constant.
Common difference: The constant difference between consecutive terms in an arithmetic sequence, usually written d.
nth term: The term in position n of a sequence, often written a_n.
Geometric sequence: A sequence where each term is found by multiplying the previous term by the same number.
Square numbers: The sequence 1, 4, 9, 16, 25, ... given by n^2.
Cube numbers: The sequence 1, 8, 27, 64, 125, ... given by n^3.
Triangular numbers: The sequence 1, 3, 6, 10, 15, ... formed by adding 1, then 2, then 3, and so on.
Fibonacci numbers: The sequence 1, 1, 2, 3, 5, 8, ... where each term is the sum of the two previous terms.
Quadratic sequence: A sequence with a constant second difference; its nth term involves n^2.
Arithmetic series: The sum of the terms of a finite arithmetic progression.

闪卡

What is an arithmetic (linear) sequence?

A sequence where the difference between each term and the one before it is always the same. This constant difference is called the common difference.

In the sequence 5, 7, 9, 11, 13, ... what is the common difference?

2, because each term is 2 more than the term before it.

What is the formula for the nth term of an arithmetic sequence?

a_n = a₁ + (n - 1)d, where a₁ is the first term, d is the common difference and a_n is the nth term.

For the sequence 5, 7, 9, 11, ..., find the 10th term using a_n = a₁ + (n - 1)d.

a₁ = 5, d = 2, so a₁₀ = 5 + (10 - 1) × 2 = 5 + 18 = 23.

How do you find the common difference of an arithmetic sequence?

Subtract any term from the term that follows it. For example, in 3, 8, 13, 18, ... the common difference is 8 - 3 = 5.

What is a term-to-term rule?

A rule that tells you how to get the next term from the previous one, such as 'add 4 each time'.

What is a position-to-term rule?

A rule that gives any term directly from its position number n, such as the nth term formula a_n = 3n + 1.

How do you decide whether a number belongs to an arithmetic sequence?

Set the nth term formula equal to that number and solve for n. If n is a positive whole number, the number is in the sequence.

Is 50 in the sequence with nth term 4n + 2?

Solve 4n + 2 = 50, giving 4n = 48 and n = 12. Since 12 is a whole number, yes, 50 is the 12th term.

What is a geometric sequence?

A sequence where each term is found by multiplying the previous term by the same fixed number, called the common ratio.

What is the common ratio of the sequence 3, 6, 12, 24, ...?

2, because each term is double the term before it (6 ÷ 3 = 2, 12 ÷ 6 = 2, and so on).

What are the square numbers?

1, 4, 9, 16, 25, 36, ... — the result of multiplying a whole number by itself, so the nth square number is n².

What are the cube numbers?

1, 8, 27, 64, 125, ... — the result of multiplying a whole number by itself twice, so the nth cube number is n³.

What are the triangular numbers?

1, 3, 6, 10, 15, 21, ... — numbers that can be arranged as a triangle of dots. The nth triangular number is n(n + 1)/2.

What is the Fibonacci sequence and its rule?

1, 1, 2, 3, 5, 8, 13, ... — each term is the sum of the two previous terms.

What is a quadratic sequence?

A sequence where the differences between terms change, but the second differences are constant. Its nth term contains an n² term.

How do you recognise a quadratic sequence from its differences?

The first differences are not constant, but the second differences (the differences of the differences) are constant.

What is a finite arithmetic progression?

A sequence with a fixed, limited number of terms that has a constant common difference, such as 2, 5, 8, 11, 14.

What is an arithmetic series?

The sum of the terms of a finite arithmetic progression, for example 2 + 5 + 8 + 11 + 14 = 40.

What is the formula for the sum of a finite arithmetic series?

S = n(a₁ + a_n)/2, where n is the number of terms, a₁ is the first term and a_n is the last term.

Use S = n(a₁ + a_n)/2 to find 2 + 5 + 8 + 11 + 14.

n = 5, a₁ = 2, a_n = 14, so S = 5 × (2 + 14)/2 = 5 × 16/2 = 40.

What is the nth term of the sequence 3, 7, 11, 15, ...?

The common difference is 4 and the first term is 3, so a_n = 4n - 1.

What is the nth term of the sequence 10, 7, 4, 1, ...?

The common difference is -3 and the first term is 10, so a_n = 13 - 3n.

How do you generate terms from a position-to-term rule?

Substitute n = 1, 2, 3, ... into the rule in turn. For example, a_n = 5n - 2 gives 3, 8, 13, 18, ...

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