Problem: A square garden has paths of increasing width. The total areas are 7, 16, 31, 52, 79 m². Find the nth term.
Step 1 & 2: Find differences
| Terms: | 7 | 16 | 31 | 52 | 79 |
| 1st differences: | 9 | 15 | 21 | 27 | |
| 2nd differences: | 6 | 6 | 6 |
Step 3: Find 'a'
a = ½ × 6 = 3, so we have 3n²
Step 4: Compare with 3n²
| n: | 1 | 2 | 3 | 4 | 5 |
| Original: | 7 | 16 | 31 | 52 | 79 |
| 3n²: | 3 | 12 | 27 | 48 | 75 |
| Difference: | 4 | 4 | 4 | 4 | 4 |
Step 5: Complete formula
The difference is constant = 4, so the formula is:
nth term = 3n² + 4
Check: When n = 4: 3(4²) + 4 = 3(16) + 4 = 48 + 4 = 52 ✓
This deep dive covers Worked Example 2: Garden Path Areas within Quadratic Sequences for GCSE Mathematics. Revise Quadratic Sequences in Algebra for GCSE Mathematics with 12 exam-style questions and 22 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 6 of 8 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Quadratic Sequences
Which of the following is a property of a quadratic sequence?
A student says: 'The sequence 3, 7, 13, 21, 31 is quadratic because the first differences increase.' Explain whether the student is correct and how to check properly.