Chapter 08 — Sequences and Series (Part 1): Introduction and Arithmetic Progressions

CHAPTER 08: SEQUENCES AND SERIES

PART 1 — INTRODUCTION AND ARITHMETIC PROGRESSIONS

A-Level Mathematics | Mathematics HipHop

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Learning Objectives

By the end of this lesson, you should be able to:

• Explain the meaning of a sequence and a series.
• Identify terms and their positions in a sequence.
• Recognise arithmetic progressions.
• Calculate the common difference of an AP.
• Find the nth term of an arithmetic progression.
• Determine missing terms and solve problems involving APs.

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1. Introduction to Sequences

A sequence is an ordered list of numbers arranged according to a particular rule or pattern.

For example:

2, 4, 6, 8, 10, ...

This sequence increases by 2 each time.

Another example is:

3, 6, 12, 24, 48, ...

Here, each term is obtained by multiplying the previous term by 2.

Sequences may be:

• Finite: having a fixed number of terms.
• Infinite: continuing without an indicated final term.

Finite sequence:

5, 10, 15, 20, 25

Infinite sequence:

5, 10, 15, 20, 25, ...

Important

The order of terms matters. A sequence is not simply a collection of numbers; each number has a definite position.

2. Terms and Their Positions

Each number in a sequence is called a term.

Consider the sequence:

4, 7, 10, 13, 16, ...

Position	1	2	3	4	5
Term	4	7	10	13	16

The first term is 4, the second term is 7, and the fifth term is 16.

We commonly represent the terms of a sequence using:

• a or u₁ for the first term.
• u₂ for the second term.
• u₃ for the third term.
• uₙ for the nth term.

The symbol uₙ represents the term in position n. It is also called the general term of a sequence.

Worked Example 1

Given the sequence 6, 10, 14, 18, 22, ...:

(a) State the first term.

(b) State the fourth term.

(c) Write the first five terms using u-notation.

Solution

(a) First term = 6.

(b) Fourth term = 18.

(c) u₁ = 6, u₂ = 10, u₃ = 14, u₄ = 18, u₅ = 22.

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3. What Is a Series?

A series is formed when the terms of a sequence are added together.

For example, consider the sequence:

2, 4, 6, 8, 10

The corresponding series is:

2 + 4 + 6 + 8 + 10

Notice the difference:

• A sequence is an ordered list of terms.
• A series is the sum of those terms.

In this chapter, we begin by studying sequences. The summation of terms will be developed further in the next parts.

4. Arithmetic Progressions (AP)

An arithmetic progression is a sequence in which the difference between any two consecutive terms is constant.

This constant difference is called the common difference, usually represented by d.

Consider:

5, 8, 11, 14, 17, ...

Subtract consecutive terms:

8 − 5 = 3

11 − 8 = 3

14 − 11 = 3

17 − 14 = 3

Since the difference is always 3, the sequence is an arithmetic progression with common difference d = 3.

Key condition for an AP

Consecutive terms must have the same difference.

d = second term − first term

5. Positive, Negative and Zero Common Differences

The common difference determines how the terms change.

Positive common difference

3, 7, 11, 15, 19, ...

d = 4. The terms increase.

Negative common difference

20, 16, 12, 8, 4, ...

d = −4. The terms decrease.

Zero common difference

6, 6, 6, 6, 6, ...

d = 0. All terms are equal.

Worked Example 2

Determine whether each of the following is an arithmetic progression. If it is, state its common difference.

(a) 7, 12, 17, 22, 27, ...

(b) 30, 25, 20, 15, 10, ...

(c) 2, 4, 8, 16, 32, ...

Solution

(a) Differences are 5, 5, 5, 5.

It is an AP with d = 5.

(b) Differences are −5, −5, −5, −5.

It is an AP with d = −5.

(c) Differences are 2, 4, 8, 16.

The differences are not constant.

It is not an AP.

6. The General Form of an Arithmetic Progression

Let:

• a = first term.
• d = common difference.

An arithmetic progression can be written as:

a, a + d, a + 2d, a + 3d, ...

Notice how each new term is obtained by adding d to the previous term.

For example, if a = 4 and d = 3, the AP is:

4, 7, 10, 13, 16, ...

7. Deriving the nth Term of an AP

The nth term is the term occupying position n in the sequence.

Consider the general AP:

a, a + d, a + 2d, a + 3d, ...

Position	Term
1	a
2	a + d
3	a + 2d
4	a + 3d
5	a + 4d
n	a + (n − 1)d

Observe the pattern:

• The first term has zero additions of d.
• The second term has one addition of d.
• The third term has two additions of d.
• The nth term has (n − 1) additions of d.

Therefore, the general formula is:

THE nth TERM OF AN AP

uₙ = a + (n − 1)d

This formula is one of the most important results in arithmetic progressions.

8. Applying the nth-Term Formula

Worked Example 3

Find the 20th term of the AP:

3, 7, 11, 15, ...

Solution

First term, a = 3.

Common difference, d = 7 − 3 = 4.

We need the 20th term, so n = 20.

Using uₙ = a + (n − 1)d:

u₂₀ = 3 + (20 − 1)(4)

= 3 + 19 × 4

= 3 + 76

u₂₀ = 79

Worked Example 4

Find the nth term of the sequence:

12, 9, 6, 3, 0, ...

Solution

First term, a = 12.

Common difference, d = 9 − 12 = −3.

Using the formula:

uₙ = a + (n − 1)d

uₙ = 12 + (n − 1)(−3)

= 12 − 3n + 3

uₙ = 15 − 3n

Worked Example 5

The nth term of an AP is given by uₙ = 5n − 2.

(a) Find the first term.

(b) Find the common difference.

(c) Find the 15th term.

Solution

(a) Put n = 1:

u₁ = 5(1) − 2 = 3.

First term = 3.

(b) Find the second term:

u₂ = 5(2) − 2 = 8.

Common difference = 8 − 3 = 5.

(c) Put n = 15:

u₁₅ = 5(15) − 2 = 73.

15th term = 73.

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9. Finding Missing Terms in an AP

Sometimes, some terms in an arithmetic progression are missing. We can use the constant difference to find them.

Worked Example 6

Find the missing terms in:

4, __, __, 16, 20, ...

Solution

The first term is 4.

The fifth term is 20.

There are four equal steps between the first and fifth terms.

Therefore:

4d = 20 − 4

4d = 16

d = 4.

The sequence is:

4, 8, 12, 16, 20, ...

Worked Example 7

The third term of an AP is 11 and the seventh term is 27. Find the first term and common difference.

Solution

Using uₙ = a + (n − 1)d:

Third term:

a + 2d = 11   (1)

Seventh term:

a + 6d = 27   (2)

Subtract equation (1) from equation (2):

4d = 16

d = 4.

Substitute into equation (1):

a + 2(4) = 11

a + 8 = 11

a = 3.

First term = 3; common difference = 4.

10. Determining Whether a Given Number Is a Term

To determine whether a number belongs to an AP, equate the nth-term formula to that number and solve for n.

If n is a positive whole number, the number is a term of the sequence.

Worked Example 8

Determine whether 101 is a term of the AP:

5, 9, 13, 17, ...

Solution

a = 5 and d = 4.

uₙ = 5 + (n − 1)4

= 5 + 4n − 4

= 4n + 1.

Set the nth term equal to 101:

4n + 1 = 101

4n = 100

n = 25.

Since 25 is a positive whole number, 101 is a term of the sequence.

Answer: Yes, 101 is the 25th term.

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11. Common Mistakes to Avoid

• Confusing a sequence with a series.
• Subtracting terms in the wrong order when finding the common difference.
• Forgetting that the first term is a, not a + d.
• Using nd instead of (n − 1)d in the nth-term formula.
• Ignoring a negative common difference.
• Accepting a fractional or negative value of n when checking whether a number is a term of an ordinary sequence indexed by positive integers.

12. Practice Exercises

Attempt all the questions before opening the answers. Show your working clearly.

Question 1

Explain the difference between a sequence and a series. Give one example of each.

Question 2

Determine whether each sequence is an AP. If it is, find the common difference.

(a) 4, 9, 14, 19, 24, ...

(b) 25, 20, 15, 10, 5, ...

(c) 1, 2, 4, 8, 16, ...

(d) −3, −3, −3, −3, ...

Question 3

Find the nth term of each AP:

(a) 6, 10, 14, 18, ...

(b) 18, 13, 8, 3, ...

(c) −2, 1, 4, 7, ...

Question 4

Find the 30th term of the AP:

7, 12, 17, 22, ...

Question 5

The nth term of an AP is uₙ = 7 − 3n.

(a) Find the first term.

(b) Find the common difference.

(c) Find the 20th term.

Question 6

The fourth term of an AP is 17 and the ninth term is 37. Find its first term and common difference.

Question 7

Find the missing terms:

−4, __, __, __, 12, ...

Question 8

Determine whether 150 is a term of the AP:

6, 12, 18, 24, ...

Question 9

The first term of an AP is 8 and the common difference is 6. Which term is 98?

Question 10

The second term of an AP is 9 and the fifth term is 21. Find the nth-term formula.

Open Answers and Worked Solutions

Question 1

A sequence is an ordered list of numbers, for example 2, 4, 6, 8.

A series is the sum of the terms, for example 2 + 4 + 6 + 8.

Question 2

(a) Differences are all 5. AP with d = 5.

(b) Differences are all −5. AP with d = −5.

(c) Differences are 1, 2, 4, 8. Not an AP.

(d) Differences are all 0. AP with d = 0.

Question 3

(a) a = 6, d = 4.

uₙ = 6 + 4(n − 1) = 4n + 2.

(b) a = 18, d = −5.

uₙ = 18 − 5(n − 1) = 23 − 5n.

(c) a = −2, d = 3.

uₙ = −2 + 3(n − 1) = 3n − 5.

Question 4

a = 7, d = 5.

u₃₀ = 7 + 29(5) = 7 + 145.

Answer: 152.

Question 5

uₙ = 7 − 3n.

(a) u₁ = 7 − 3 = 4.

(b) u₂ = 7 − 6 = 1, so d = 1 − 4 = −3.

(c) u₂₀ = 7 − 60 = −53.

Question 6

a + 3d = 17

a + 8d = 37

Subtracting gives 5d = 20, so d = 4.

a + 12 = 17, so a = 5.

Answer: a = 5, d = 4.

Question 7

There are four equal intervals from −4 to 12.

4d = 16, so d = 4.

Answer: −4, 0, 4, 8, 12, ...

Question 8

uₙ = 6 + (n − 1)6 = 6n.

6n = 150 gives n = 25.

Answer: Yes, 150 is the 25th term.

Question 9

98 = 8 + (n − 1)6.

90 = 6(n − 1).

15 = n − 1, so n = 16.

Answer: 98 is the 16th term.

Question 10

Second term: a + d = 9.

Fifth term: a + 4d = 21.

Subtracting gives 3d = 12, so d = 4.

a + 4 = 9, so a = 5.

uₙ = 5 + 4(n − 1).

Answer: uₙ = 4n + 1.

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13. Lesson Summary

In this lesson, we have established that:

• A sequence is an ordered list of numbers.
• A series is the sum of terms of a sequence.
• An arithmetic progression has a constant difference between consecutive terms.
• The common difference is represented by d.
• The first term is represented by a.
• The nth term of an AP is uₙ = a + (n − 1)d.

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