Pattern: 1, 4, 9, 16, __
Question
Look at this number pattern: 1, 4, 9, 16, ___
What is the next number? Identify the pattern and write the next three terms.
Solution — Step by Step
Step 1: Look at the relationship between consecutive terms.
- 1 to 4: difference of 3
- 4 to 9: difference of 5
- 9 to 16: difference of 7
The differences are 3, 5, 7 — consecutive odd numbers. The next difference should be 9.
So the next term = 16 + 9 = 25
Step 2: Look for a deeper pattern — what are these numbers?
Let us check if each number is a perfect square:
- 1 = 1 × 1 = 1²
- 4 = 2 × 2 = 2²
- 9 = 3 × 3 = 3²
- 16 = 4 × 4 = 4²
- Next: 5 × 5 = 5² = 25 ✓
This confirms: these are square numbers (perfect squares)!
Step 3: Write the next three terms.
The sequence continues:
- 5th term: 5² = 25
- 6th term: 6² = 36
- 7th term: 7² = 49
Answer: The next three terms are 25, 36, 49.
The pattern is the sequence of perfect squares: 1², 2², 3², 4², 5², 6², 7², …
Square Numbers (Perfect Squares)
n-th term of this pattern = n²
1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49 8² = 64 9² = 81 10² = 100
Why This Works
A square number is what we get when we multiply a whole number by itself. The name "square" comes from the fact that these numbers can be arranged in a perfect square shape.
Imagine arranging small tiles:
- 1 tile → 1×1 square ← 1
- 4 tiles → 2×2 square ← 4
- 9 tiles → 3×3 square ← 9
- 16 tiles → 4×4 square ← 16
Each term is a square of tiles. This physical arrangement is why these are called square numbers.
📌 Note
There is a beautiful pattern in the differences between consecutive square numbers: 4 − 1 = 3 9 − 4 = 5 16 − 9 = 7 25 − 16 = 9 36 − 25 = 11
The differences are always odd numbers: 3, 5, 7, 9, 11, … This is a great shortcut to find the next square number without multiplying!
The Odd Number Shortcut
We can build the square number sequence by adding successive odd numbers:
1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25
So if you know the current square number, just add the next odd number to get the next square number. This is a quick mental maths trick!
Common Mistake
⚠️ Common Mistake
Mistake: Saying the pattern is "add 5 each time" because 4 − 1 = 3 is ignored and students only check 9 − 4 = 5 and 16 − 9 = 7... actually the differences are increasing, not constant.
Or sometimes students see 1, 4, 9, 16 and write the next term as 25 but cannot explain why. In exams, always describe the pattern: "these are the squares of 1, 2, 3, 4, so the next is 5² = 25."
🎯 Exam Insider
Pattern questions almost always ask you to identify AND explain the pattern. Write: "The given pattern consists of the squares of natural numbers (1², 2², 3², 4²...). The next term is 5² = 25, followed by 6² = 36 and 7² = 49."