Sequences and series: patterns, terms and notation

$2, 4, 6, 8, 10, \ldots$ — a list of even numbers. $1, 3, 9, 27, 81, \ldots$ — each number is three times the last. $1, 1, 2, 3, 5, 8, 13, \ldots$ — each number is the sum of the previous two. These are all sequences — ordered lists of numbers that follow a pattern.

When you add up the terms of a sequence, you get a series. The sum $2 + 4 + 6 + 8 + 10 = 30$ is a series formed from the first five terms of the sequence above.

Sequences and series appear throughout WAEC mathematics — in arithmetic progressions, geometric progressions, compound interest, biological growth, payment plans, and more. This topic teaches you the two most important kinds (AP and GP), how to find any term, and how to sum them up.

This lesson introduces the vocabulary and notation. The next lesson covers APs and GPs in detail: the $n$-th term, the sum of $n$ terms and the sum to infinity.

Learning objectives

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

• define a sequence and a series
• describe a sequence by listing, by a recurrence rule and by a formula for the $n$-th term
• recognise arithmetic and geometric sequences
• use sigma notation for a series

How to describe a sequence

A sequence can be described in three ways:

1. List the terms

$2, 4, 6, 8, 10, \ldots$

For short sequences, just listing the terms is fine. But for the $100$th term, listing all terms is impractical.

2. Recurrence — describe each term in terms of the previous

$a_1 = 2$, and $a_n = a_{n-1} + 2$ for $n \geq 2$.

This says: "The first term is $2$, and each subsequent term is the previous one plus $2$."

Useful for understanding the pattern, but slow if you want term $1000$.

3. Formula — describe each term directly in terms of $n$

$a_n = 2n$.

Now you can compute any term: $a_5 = 2 \times 5 = 10$. $a_{100} = 2 \times 100 = 200$.

The formula form is the most powerful, and it's what most WAEC questions ask you to find.

Patterns to look for

When you see a sequence, ask yourself: what's the pattern? Here are the most common.

Constant difference (arithmetic)

$3, 7, 11, 15, 19, \ldots$ — each term is $4$ more than the previous.

The difference between consecutive terms is constant. This is an arithmetic progression (AP) or arithmetic sequence.

Constant ratio (geometric)

$2, 6, 18, 54, 162, \ldots$ — each term is $3$ times the previous.

The ratio of consecutive terms is constant. This is a geometric progression (GP) or geometric sequence.

Squares, cubes, etc.

$1, 4, 9, 16, 25, \ldots$ — squares: $a_n = n^2$.

$1, 8, 27, 64, 125, \ldots$ — cubes: $a_n = n^3$.

WAEC sometimes presents these patterns directly.

Fibonacci pattern

$1, 1, 2, 3, 5, 8, 13, 21, \ldots$ — each term is the sum of the previous two: $a_n = a_{n-1} + a_{n-2}$.

Less often tested, but appears in some advanced WAEC questions.

Figure 1. An AP adds the same number each time; a GP multiplies by the same number each time. WASSCE Tutor diagram.

Identifying arithmetic and geometric sequences

Quick test for AP

Subtract each term from the next. If the differences are all the same, it's an AP. The common difference is denoted $d$.

Example: $3, 7, 11, 15$.

• $7 - 3 = 4$
• $11 - 7 = 4$
• $15 - 11 = 4$ All differences are $4$. This is an AP with first term $a = 3$ and common difference $d = 4$.

Quick test for GP

Divide each term by the previous. If the ratios are all the same, it's a GP. The common ratio is denoted $r$.

Example: $2, 6, 18, 54$.

• $6 / 2 = 3$
• $18 / 6 = 3$
• $54 / 18 = 3$ All ratios are $3$. This is a GP with first term $a = 2$ and common ratio $r = 3$.

Combined example

Example: $1, 2, 4, 7, 11, 16, \ldots$

• Differences: $1, 2, 3, 4, 5, \ldots$ (the differences themselves form a sequence!)
• Ratios: $2, 2, 1.75, \ldots$ — not constant.

This is neither AP nor GP. The differences form an AP — this is called a quadratic sequence in some contexts. WAEC focuses on plain APs and GPs.

Series — adding up the terms

Finite sum

The sum of the first $n$ terms is $S_n$.

Example: For the AP $3, 7, 11, 15, \ldots$, the first four terms sum to: $S_4 = 3 + 7 + 11 + 15 = 36$.

Example: For the GP $2, 6, 18, 54$, the sum is: $S_4 = 2 + 6 + 18 + 54 = 80$.

WAEC tests formulas for $S_n$ for both APs and GPs (see the next lesson).

Infinite sum (only for some GPs)

For an infinite GP, the sum may converge to a finite limit if $|r| < 1$. Otherwise it diverges to infinity.

Example: $1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \ldots = 2$ (this converges).

Example: $1 + 2 + 4 + 8 + 16 + \ldots = \infty$ (this diverges).

We cover infinite GP sums in the next lesson.

Notation conventions

• $a_n$ or $T_n$ — the $n$-th term of a sequence. WAEC often uses $T_n$.
• $a$ — the first term.
• $d$ — common difference (for AP).
• $r$ — common ratio (for GP).
• $S_n$ — sum of the first $n$ terms.
• $S_\infty$ — infinite sum (if it converges).

Examples — recognizing the type

Example 1

A bakery sells $50$ loaves on the first day, then increases by $5$ loaves each subsequent day.

This is an AP with $a = 50$, $d = 5$.

Terms: $50, 55, 60, 65, 70, \ldots$

Example 2

An investment of $1000$ NLe grows by $10\%$ each year.

This is a GP with $a = 1000$, $r = 1.10$.

Terms: $1000, 1100, 1210, 1331, \ldots$

Example 3

The population of a village in Sierra Leone is $1000$ this year. It declines by $50$ people each year.

This is an AP with $a = 1000$, $d = -50$.

Terms: $1000, 950, 900, 850, \ldots$

Example 4

A car loses $25\%$ of its value each year. Initial value $5000$ NLe.

This is a GP with $a = 5000$, $r = 0.75$.

Terms: $5000, 3750, 2812.50, \ldots$

When the sequence is given in non-standard form

Sometimes WAEC presents a sequence by listing terms (not telling you which kind). You need to identify the type yourself.

Example: The sequence $3, 9, 27, 81, \ldots$ is given. Is it AP or GP?

Differences: $6, 18, 54$ — not constant. So not AP. Ratios: $3, 3, 3$ — constant. GP with $a = 3, r = 3$.

Alternatively, the formula is $a_n = 3^n$.

Example: The sequence $4, 7, 12, 19, 28, \ldots$ is given.

Differences: $3, 5, 7, 9$ — not constant. The differences themselves form an AP. Ratios: $7/4, 12/7, \ldots$ — not constant.

This is neither AP nor GP. It's a quadratic-pattern sequence.

The connection to formulas

For sequences with a clear pattern, you can write a formula.

Example: $1, 4, 9, 16, 25, \ldots$ → $a_n = n^2$.

Example: $2, 4, 6, 8, 10, \ldots$ → $a_n = 2n$.

Example: $3, 6, 12, 24, 48, \ldots$ → $a_n = 3 \times 2^{n-1}$ (GP with $a = 3, r = 2$).

The next lesson derives the general formulas for AP and GP.

WASSCE examination focus

Practice problems

For each sequence, identify whether it is AP, GP, both, or neither. If it is AP, give the common difference. If it is GP, give the common ratio.

1. $4, 9, 14, 19, 24$

2. $1, 5, 25, 125$

3. $8, 4, 2, 1$

4. $-2, 1, 4, 7, 10$

5. $3, 3, 3, 3, 3$

6. $1, 2, 4, 7, 11$

7. $-1, 3, -9, 27, -81$

8. $100, 95, 90, 85$

9. Find the next two terms of $5, 8, 11, 14, \ldots$

10. Find the next two terms of $\dfrac{1}{2}, 1, 2, 4, \ldots$

11. A small business sells $100$ units in month $1$. Each month, sales grow by $10\%$. Identify the type of sequence.

12. Aminata saves $20$ NLe in week $1$, then $5$ NLe more each subsequent week. Identify the type.

Worked solutions

1. Differences: $5, 5, 5, 5$. AP, $d = 5$.

2. Ratios: $5, 5, 5$. GP, $r = 5$.

3. Ratios: $\dfrac{1}{2}, \dfrac{1}{2}, \dfrac{1}{2}$. GP, $r = \dfrac{1}{2}$. (Differences: $-4, -2, -1$ — not constant.)

4. Differences: $3, 3, 3, 3$. AP, $d = 3$.

5. Constant sequence. Both AP ($d = 0$) and GP ($r = 1$).

6. Differences: $1, 2, 3, 4$ — not constant. Ratios: $2, 2, 1.75, \ldots$ — not constant. Neither.

7. Ratios: $-3, -3, -3, -3$. GP, $r = -3$.

8. Differences: $-5, -5, -5$. AP, $d = -5$.

9. AP with $a = 5, d = 3$. Next two: $17, 20$.

10. GP with $a = \dfrac{1}{2}, r = 2$. Next two: $8, 16$.

11. GP (constant percentage growth = constant ratio).

12. AP (constant addition each week).

What's next

In the next lesson we go deep on arithmetic and geometric progressions: the general formula for the $n$-th term, the sum of the first $n$ terms, the sum to infinity of a GP, and the WAEC questions built on them.