Grade 12 Mathematics · Unit 1 · Chapter 1.2 · QuickNotes

Arithmetic and Geometric Sequences

These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 12 Mathematics Textbook, Unit 1, Chapter 1.2.

~7 min read
Summary
  • An arithmetic sequence grows by adding the same number d each time, and its nth term is An = A1 + (n − 1)d.
  • A geometric sequence grows by multiplying by the same number r each time, and its nth term is Gn = G1 · rn−1.
  • The arithmetic mean of two numbers is their average, while the geometric mean of two numbers is the square root of their product.

Key Words

  • Arithmetic sequence: an arithmetic sequence is one where each term is found by adding a fixed number to the term before it.
  • Common difference (d): the common difference is the fixed number that you add each time in an arithmetic sequence.
  • Arithmetic mean: the arithmetic mean of two numbers is the middle term that sits between them in an arithmetic sequence.
  • Geometric sequence: a geometric sequence is one where each term is found by multiplying the term before it by a fixed number.
  • Common ratio (r): the common ratio is the fixed number that you multiply by each time in a geometric sequence, and it is never zero.
  • Geometric mean: the geometric mean of two numbers is the middle term that sits between them in a geometric sequence.

What This Chapter Is About

  • This chapter teaches you the two most important kinds of sequences, the arithmetic kind and the geometric kind.
  • You learn how to find any term with a formula and how to find the mean that sits between two numbers in each kind.
1 2 3 4 5 n 2 5 8 11 14 Arithmetic: add 3 each time 1 2 3 4 5 n 3 12 24 36 48 Geometric: multiply by 2 each time

Figure 1.2: an arithmetic sequence rises in a straight line, while a geometric sequence curves upward quickly.

Arithmetic Sequences

  • In an arithmetic sequence, you get the next term by adding the same fixed number to the term before it.
  • This fixed number is called the common difference d, and you find it by subtracting any term from the term right after it.
  • The common difference can be positive, which makes the sequence rise, or negative, which makes it fall.
  • To check if a sequence is arithmetic, test whether the difference between each pair of neighbouring terms stays the same.

An = A1 + (n − 1)d

This means you start at the first term A1 and add the common difference d a total of (n − 1) times to reach the nth term.

Worked example: find the 6th term of 2, 5, 8, 11, …

Here A1 = 2 and the common difference is d = 3. So A6 = A1 + (6 − 1)d = 2 + 5(3) = 2 + 15 = 17.

Arithmetic Mean Between Two Numbers

  • The arithmetic mean is the term that sits exactly between two numbers in an arithmetic sequence.
  • Because the differences are equal, the middle term m of a, m, b is just the average, so m = (a + b) ÷ 2.

Worked example: find the arithmetic mean of 1 and 8.

The mean is m = (1 + 8) ÷ 2 = 9 ÷ 2 = 4.5, so 1, 4.5, 8 is an arithmetic sequence.

Geometric Sequences

  • In a geometric sequence, you get the next term by multiplying the term before it by the same fixed number.
  • This fixed number is called the common ratio r, and you find it by dividing any term by the term right before it.
  • The common ratio is never zero, but it can be a fraction, which makes the terms shrink, or a negative number, which makes the signs flip.

Gn = G1 · rn−1

This means you start at the first term G1 and multiply by the common ratio r a total of (n − 1) times to reach the nth term.

Worked example: find the 6th term of 3, 6, 12, 24, …

Here G1 = 3 and the common ratio is r = 2. So G6 = G1 · r5 = 3 × 25 = 3 × 32 = 96.

Geometric Mean Between Two Numbers

  • The geometric mean is the term that sits between two numbers in a geometric sequence.
  • Because the ratios are equal, the middle term m of a, m, b satisfies m2 = ab, so m = ±√(ab).
  • You keep both the positive and the negative answer, unless the question limits you to one sign.

Worked example: find the geometric mean of 2 and 8.

Use m2 = ab = 2 × 8 = 16, so m = ±√16 = ±4. The geometric mean is 4 or −4.

Quick Comparison Table

Feature Arithmetic sequence Geometric sequence
How you move to the next term Add the common difference d Multiply by the common ratio r
How to find the constant Subtract neighbouring terms Divide neighbouring terms
Formula for the nth term An = A1 + (n − 1)d Gn = G1 · rn−1
Middle term of a, m, b m = (a + b) ÷ 2 m = ±√(ab)

Common Mistakes to Avoid

  • Do not add for a geometric sequence or multiply for an arithmetic sequence, because each kind uses only one operation.
  • Do not use n in the formula where it should be (n − 1), since you add d or multiply by r one fewer time than the term number.
  • Do not drop the ± sign on the geometric mean, because both the positive and the negative root can be correct.
  • Do not let the common ratio be zero, because a zero ratio would make every term after the first equal to zero.

Easy Ways to Remember

  • Arithmetic means “add”, while geometric makes the terms “grow by times”, so think add for one and multiply for the other.
  • For the nth term, both formulas use (n − 1), so you take one step fewer than the term number.
  • The arithmetic mean is the average with a plus, and the geometric mean is the square root with a times.

Quiz

Tap an answer to check it.

1. What is the common difference of 7, 11, 15, 19, …?

2. Find the 10th term of the arithmetic sequence with A1 = 3 and d = 5.

3. What is the common ratio of 2, 6, 18, 54, …?

4. Find the 5th term of the geometric sequence with G1 = 1 and r = 2.

5. What is the geometric mean of 3 and 12?

Remember: An arithmetic sequence adds the common difference d, so An = A1 + (n − 1)d. A geometric sequence multiplies by the common ratio r, so Gn = G1 · rn−1. The arithmetic mean is the average, and the geometric mean is ±√(ab).

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