Number System Fundamentals • Topic 5 of 7

Number Series

A number series follows a rule: add a constant (arithmetic), multiply by a constant (geometric), add growing amounts (squares, cubes, differences of differences), or alternate two patterns. The first move is always to compute the differences between consecutive terms. A constant difference signals an arithmetic series; a constant ratio signals a geometric one. The sum of an arithmetic series is (number of terms) × (first + last) / 2. Two handy results: the sum of the first n odd numbers is n², and the sum of the first n even numbers is n(n+1). Recognising the pattern is the whole game.

✅ Solved examples

1. Find the next term: 2, 5, 8, 11, …
The differences are all 3, so this is arithmetic. The next term is 11 + 3 = 14.
2. Find the next term: 3, 6, 12, 24, …
Each term is double the previous one, so this is geometric with ratio 2. The next term is 24 × 2 = 48.
3. What is the sum of the first 20 odd numbers?
The sum of the first n odd numbers is n². With n = 20, the sum is 20² = 400.
4. Find the next term: 1, 4, 9, 16, …
These are the perfect squares 1², 2², 3², 4². The next term is 5² = 25.

✏️ Practice — try these, take hints as needed

1. Find the next term: 5, 10, 20, 40, …
Check the ratio between terms.
Each term is twice the one before.
Double 40.
80.
2. Find the next term: 2, 6, 12, 20, 30, …
Compute the differences: 4, 6, 8, 10.
The differences increase by 2 each time.
The next difference is 12, so add it to 30.
42.
3. What is the sum of the first 15 natural numbers?
Use n(n+1)/2.
Substitute n = 15.
Compute 15 × 16 / 2.
120.
4. Find the next term: 1, 1, 2, 3, 5, 8, …
Look at how each term relates to the two before it.
Each term is the sum of the previous two (Fibonacci).
Add 5 + 8.
13.
5. What is the sum of the first 10 even numbers?
The sum of the first n even numbers is n(n+1).
Substitute n = 10.
Compute 10 × 11.
110.

📝 Topic test — 8 questions

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