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 7 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.
Topic position
Section 6 of 7
Practice
12 questions
Recall
22 flashcards
Worked Example 2: Garden Path Areas
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 ✓