Sequences and the nth term

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Apuntes de la lección

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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Preguntas de práctica

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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

Más preguntas

9.Match each sequence to its term-to-term rule.
Easy
- 3, 6, 12, 24, ...
- 10, 7, 4, 1, ...
- 1, 4, 9, 16, ...
- 1, 1, 2, 3, 5, ...
- Multiply by 2
- Subtract 3
- Add the two previous terms
- Add consecutive odd numbers
10.Put these sequences in order of their common difference, from smallest to largest.
Medium
- 5, 8, 11, 14, ...
- 20, 18, 16, 14, ...
- 2, 4, 6, 8, ...
- 1, 6, 11, 16, ...
11.Which of the following sequences are geometric? (select all that apply)
Medium
- A2, 6, 18, 54, ...
- B1, 4, 9, 16, ...
- C80, 40, 20, 10, ...
- D3, 6, 9, 12, ...
- E1, 2, 4, 8, ...
12.The first term of an arithmetic sequence is 4 and the common difference is 3. Which expression gives the nth term?
Medium
- A3n + 1
- B4n + 3
- C3n + 4
- D4n - 1
13.Which of these numbers is a term of the sequence with nth term 3n + 1?
Medium
- A18
- B20
- C22
- D25
14.The sequence 1, 1, 2, 3, 5, 8, ... is called the Fibonacci sequence.
Easy
True or false?
15.Which of the following are special sequences? (select all that apply)
Medium
- ASquare numbers
- BFibonacci numbers
- CPrime numbers
- DCube numbers
- EEven numbers
16.Match each sequence to its name.
Medium
- 1, 4, 9, 16, ...
- 1, 8, 27, 64, ...
- 1, 3, 6, 10, ...
- 1, 1, 2, 3, 5, ...
- Square numbers
- Cube numbers
- Triangular numbers
- Fibonacci numbers
17.Put these steps in the correct order to find the nth term of the arithmetic sequence 5, 9, 13, 17, ...
Medium
- Write the nth term as 4n + 1
- Find the common difference (4)
- Check the first term: 4×1 + 1 = 5
- Multiply the term number by the common difference: 4n
18.A sequence starts 3, 8, 13, 18, ... What is the common difference?
Easy
- A5
- B3
- C8
- D4
19.A sequence has first term 6 and common difference 4. What is the 7th term?
Medium
- A30
- B34
- C28
- D24
20.The nth term of a sequence is 6n - 5. What is the 9th term?
Medium
- A49
- B54
- C43
- D59
21.In the sequence 4, 12, 20, 28, ... each term is found by adding 8 to the previous term.
Easy
True or false?
22.Which of these numbers are terms of the sequence with nth term 2n + 7? (select all that apply)
Medium
- A13
- B21
- C19
- D24
- E17
23.Put these steps in the correct order to find the nth term of the arithmetic sequence 6, 10, 14, 18, ...
Medium
- Write the common difference as the coefficient of n: 4n
- Find the common difference by subtracting successive terms: 4
- Check the formula with the first term: 4 x 1 + 2 = 6
- Find the adjustment by comparing 4n with the first term: 4 x 1 = 4, so add 2
24.Which expression gives the nth term of the sequence?
Medium
- A5n + 4
- B5n + 9
- C9n + 5
- D5n - 4
25.The number 104 is a term of this sequence.
Easy
True or false?
26.A sequence starts 4, 10, 16, 22, ... Which expression gives the nth term?
Medium
- A6n - 2
- B6n + 4
- C4n + 6
- D6n
27.A sequence has nth term 3n + 2. What is the 12th term?
Medium
- A38
- B36
- C40
- D34
28.Which of these numbers is a term of the sequence with nth term 4n - 1?
Hard
- A39
- B40
- C41
- D42
29.The sequence 2, 6, 10, 14, ... is arithmetic with common difference 4.
Easy
True or false?
30.In the sequence 3, 7, 11, 15, ..., what is the common difference?
Easy
- A4
- B3
- C7
- D11
31.A sequence has nth term 5n + 2. What is the 6th term?
Easy
- A32
- B30
- C27
- D35
32.The sequence 2, 4, 8, 16, ... is geometric with common ratio 2.
Easy
True or false?
33.Which of these statements about an arithmetic sequence are true? (select all that apply)
Medium
- AThe difference between consecutive terms is constant.
- BThe nth term can be written in the form a₁ + (n - 1)d.
- CEach term is found by multiplying the previous term by a fixed number.
- DThe graph of the terms lies on a straight line.
- EThe common difference is always positive.
34.Match each sequence to its common difference.
Medium
- 3, 8, 13, 18, ...
- 20, 17, 14, 11, ...
- 1, 4, 7, 10, ...
- 6, 10, 14, 18, ...
- 5
- -3
- 4
35.Put these steps in the correct order to find the nth term of the arithmetic sequence 4, 9, 14, 19, ...
Medium
- Write the nth term as 5n - 1.
- Find the common difference: 9 - 4 = 5.
- Check: when n = 1, 5(1) - 1 = 4.
- The coefficient of n is 5; the sequence is 5n plus or minus a constant.
36.The nth term of a sequence is 3n² - 1. What is the 4th term?
Medium
- A47
- B48
- C35
- D11
37.An arithmetic sequence has first term a₁ = 6 and common difference d = -4. What is the 10th term?
Hard
- A-30
- B-34
- C42
- D30
38.Which of the following statements about the arithmetic sequence 5, 7, 9, 11, 13, 15, ... are true? (select all that apply)
Medium
- AThe common difference is 2.
- BThe 10th term is 23.
- CThe sequence is decreasing.
- DThe 1st term is 5.
39.The first term of an arithmetic sequence is 4 and the 6th term is 19. What is the 10th term?
Hard
- A31
- B34
- C37
- D40
40.Which of the following numbers are terms of the sequence with nth term 5n - 2? (select all that apply)
Hard
- A3
- B13
- C20
- D23
- E28
41.An arithmetic sequence starts 4, 9, 14, 19, ... Find the 20th term.
Medium
42.Complete the sentence about the formula for the nth term of an arithmetic sequence.
Easy
For an arithmetic sequence, the nth term is given by a____n = a___₁ + (n - 1)d, where a___₁ is the first term and d is the ____.
43.Name the special sequence described.
Easy
The sequence 1, 4, 9, 16, 25, ... is called the ____ numbers.
44.The first term of an arithmetic sequence is 3 and the common difference is 4. Find the 15th term.
Medium
45.An arithmetic sequence begins 11, 15, 19, 23, ... Find the 12th term.
Medium
46.Complete the sentence about the position-to-term rule for an arithmetic sequence.
Medium
For an arithmetic sequence with first term a___₁ and common difference d, the nth term is given by a____n = ____ .
47.A sequence has nth term 5n - 4. Work out the 11th term.
Medium
48.What name is given to a sequence in which each term after the first is found by multiplying the previous term by the same number?
Easy
49.Find the 10th term of the sequence.
Medium
50.An arithmetic sequence has first term a₁ = 7 and common difference d = 5. Work out the 8th term.
Easy
51.The nth term of a sequence is 9n - 4. Work out the 13th term.
Medium
52.An arithmetic sequence begins 8, 15, 22, 29, ... Work out the 25th term.
Medium
53.Explain how you can tell whether a number is a term of the sequence with nth term 5n + 2.
Hard
54.Explain how you can tell whether a number is a term of the sequence with nth term 4n + 3.
Medium
55.The nth term of a sequence is 7n - 3. Explain how you would decide whether 88 is a term of this sequence.
Hard
56.The nth term of a sequence is 8n - 5. Explain how you would decide whether 123 is a term of this sequence.
Hard
57.A sequence has nth term n² + 2n. Explain how you can tell that the sequence is not arithmetic.
Hard
58.An arithmetic sequence starts 4, 9, 14, 19, ... Find the 20th term.
Medium
- A119
- B99
- C109
- D89
59.The first term of an arithmetic sequence is 3 and the common difference is 4. Find the 15th term.
Medium
- A65
- B53
- C59
- D71
60.An arithmetic sequence begins 11, 15, 19, 23, ... Find the 12th term.
Medium
- A67
- B49
- C61
- D55
61.A sequence has nth term 5n - 4. Work out the 11th term.
Medium
- A56
- B51
- C46
- D61
62.Find the 10th term of the sequence.
Medium
- A59
- B54
- C49
- D64
63.An arithmetic sequence has first term a₁ = 7 and common difference d = 5. Work out the 8th term.
Easy
- A38
- B42
- C50
- D46
64.The nth term of a sequence is 9n - 4. Work out the 13th term.
Medium
- A135
- B113
- C124
- D102
65.An arithmetic sequence begins 8, 15, 22, 29, ... Work out the 25th term.
Medium
- A158
- B194
- C176
- D212

Apuntes de repaso

## 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.

Tarjetas

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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