Sequences, Letters and Position-to-Term Rules

British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course

This page is about patterns and the first ideas of algebra. You will see that a letter can stand for a number, and that a sequence such as 4, 8, 12, 16, … has a rule that lets you jump straight to the 100th term without writing out all the terms before it.

Key words

1. Letters that stand for numbers

In maths we sometimes use a letter instead of a number. The letter stands for a quantity that can vary (change). Look at a + b = 15. If a is 6, then b must be 9. If a is 10, b must be 5. The letters change, but the rule "they add up to 15" stays the same.

Every pair in the table makes a + b = 15 true. If we are told the numbers must be whole numbers, the decimal pair (7.5, 7.5) is not allowed — always read the conditions in the question.

Subtraction works in the same way. m − n = 8 means m is always 8 more than n: if n = 5, m = 13; if n = 12, m = 20.

1. Write the calculation with the number you know put in place of its letter.
2. Use the inverse operation to find the other letter: if a + b = 15 then b = 15 − a; if m − n = 8 then m = n + 8 and n = m − 8.
3. Check by putting both numbers back into the original calculation.

2. Term-to-term and position-to-term rules

Look at the sequence 4, 8, 12, 16, 20, … There are two ways to describe it:

• Term-to-term rule: start at 4 and add 4 each time. This is good for finding the next term.
• Position-to-term rule: term = position × 4. This lets you find any term straight away.

Why does this work? The 3rd term is 4 + 4 + 4. Adding 4 three times is the same as 3 × 4. In the same way, the 50th term is 4 added 50 times, which is 50 × 4 = 200. Repeated addition of the same number is multiplication.

3. Finding and using a position-to-term rule

1. Make a table with the positions 1, 2, 3, … above the terms.
2. Find the step (the difference between the terms). If it is the same every time, it is the number you multiply by.
3. Check with the first term: when the sequence is a times table, term 1 equals the step (e.g. step 7, first term 7).
4. Write the rule in words: term = position × 7.
5. Use it: to find the 20th term, do 20 × 7 = 140.
6. To go backwards (which position is the term 84?), use the inverse: 84 ÷ 7 = 12, so it is the 12th term. If the division does not give a whole number, the number is not in the sequence.

4. Patterns of shapes

Many Checkpoint questions show a pattern made of dots, sticks or tiles. Count how many are added each time — that is the multiplier.

Pattern 15 has 15 columns of 3 dots: 15 × 3 = 45 dots. Which pattern has 60 dots? 60 ÷ 3 = 20, so Pattern 20.

5. Square-number sequences

A square number is what you get when you multiply a whole number by itself: 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9 and so on. We can write 5 × 5 as 5² ("5 squared"). The name comes from the dot pattern: each square number makes a square.

In the sequence 1, 4, 9, 16, 25, … the rule is: term = position × position. So the 10th term is 10 × 10 = 100 and the 12th term is 12 × 12 = 144.

• Spot it: the differences between square numbers are the odd numbers 3, 5, 7, 9, 11, … — they do not stay the same, so it is not a times-table sequence.
• Is a number in the sequence? Find a number that multiplies by itself to make it. 81 = 9 × 9, so 81 is the 9th term. 50 is not: 7 × 7 = 49 and 8 × 8 = 64.
• Rules built on squares: "square the position, then add 1" gives 2, 5, 10, 17, 26, … You still use your square numbers first.

Worked examples

Worked example 1 — a + b = 15. (a) Find b when a = 6. (b) Find a when b = 15. [2 marks]

1. (a) Put 6 in place of a: 6 + b = 15. Inverse: b = 15 − 6 = 9. Check: 6 + 9 = 15 ✓
2. (b) a + 15 = 15, so a = 15 − 15 = 0.

Common mistake: writing 6 + 15 = 21 for (a). The total is already 15 — b is the missing part.

Worked example 2 — m − n = 8, where m and n are whole numbers less than 20. (a) Find m when n = 5. (b) Write two more pairs of values for m and n. [2 marks]

1. (a) m − 5 = 8. Inverse: m = 8 + 5 = 13. Check: 13 − 5 = 8 ✓
2. (b) Choose any n, then add 8 — but m must stay below 20, so n can be at most 11. For example n = 2, m = 10 and n = 11, m = 19.

Worked example 3 — Here is a sequence: 7, 14, 21, 28, … (a) Write the position-to-term rule. (b) Find the 20th term. [2 marks]

1. The step is +7 every time, and the 1st term is 7, so these are the multiples of 7.
2. (a) term = position × 7 (accept "multiply the position by 7").
3. (b) 20 × 7 = 140.

Common mistake: writing "add 7" for (a). That is the term-to-term rule, not the position-to-term rule.

Worked example 4 — Look at the dot patterns in section 4 (3 dots in Pattern 1, 6 in Pattern 2, 9 in Pattern 3). (a) How many dots are in Pattern 15? (b) Which pattern has exactly 60 dots? [2 marks]

1. Each pattern adds 3 dots, so dots = pattern number × 3.
2. (a) 15 × 3 = 45 dots
3. (b) inverse: 60 ÷ 3 = 20 → Pattern 20

Worked example 5 — The sequence of square numbers starts 1, 4, 9, 16, 25, … (a) What is the 10th term? (b) Is 50 a term in this sequence? Explain. [2 marks]

1. (a) term = position × position = 10 × 10 = 100
2. (b) No. 7 × 7 = 49 and 8 × 8 = 64. 50 is between them, so no whole-number position gives 50.

Worked example 6 — Sara is looking at the sequence 6, 12, 18, 24, … She says: "The 50th term is 56, because I add 6 to 50." Is she correct? Explain. [1 mark]

1. Each term is the position multiplied by 6 (6 is added again and again, which is multiplication).
2. 50th term = 50 × 6 = 300.
3. No — she should multiply, not add. The 50th term is 300. (Check her rule with the 1st term: 1 + 6 = 7, but the 1st term is 6.)

Practice

All pages in this course

Start here

• Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

• Counting On and Back and Negative Numbers
• Sequences, Letters and Position-to-Term Rules
• Order of Operations, Brackets and Laws of Arithmetic
• Multiplying Whole Numbers (up to 10 000 × 2-digit)
• Dividing Whole Numbers (up to 1000 ÷ 2-digit)
• Factors, Multiples and Tests of Divisibility
• Square Numbers and Cube Numbers

Number: place value, decimals, fractions, percentages and ratio

• Place Value and Decimals to Thousandths
• Rounding Decimals
• Fractions: Division, Operators and Simplest Form
• Adding and Subtracting Fractions
• Multiplying and Dividing Fractions by Whole Numbers
• Fractions, Decimals and Percentages: Equivalence and Ordering
• Adding, Subtracting, Multiplying and Dividing Decimals
• Ratio and Direct Proportion

Geometry and measure

• Time: Decimals and Mixed Units
• Quadrilaterals: Properties and Classification
• Circles: Parts and Constructing Circles
• Angles: Measuring, Drawing and Angles in a Triangle
• Area of Rectangles and Right-angled Triangles
• 3D Shapes, Compound Shapes and Nets
• Capacity, Volume and Surface Area
• Coordinates in Four Quadrants
• Rotational Symmetry, Translation, Reflection and Rotation

Statistics and probability

• Collecting and Representing Data
• Mode, Median, Mean and Range, and Interpreting Data
• Probability

Practice tests

• Checkpoint 6 Maths — Practice Test 1
• Checkpoint 6 Maths — Practice Test 2
• Checkpoint 6 Maths — Practice Test 3