Photo AI
Ryan is making a sequence of patterns using counters - OCR - GCSE Maths - Question 13 - 2023 - Paper 2
Question 13
Ryan is making a sequence of patterns using counters. Here are the first four patterns in the sequence. Ryan started with 80 counters. Ryan says I still have enoug... show full transcript
Worked Solution & Example Answer:Ryan is making a sequence of patterns using counters - OCR - GCSE Maths - Question 13 - 2023 - Paper 2
Step 1
Is Ryan correct?
Answer
To determine if Ryan is correct, we first need to find the number of counters required for Pattern 5 and Pattern 6.
From the provided sequence:
- Pattern 1: 3 counters
- Pattern 2: 6 counters
- Pattern 3: 10 counters
- Pattern 4: 15 counters
The pattern appears to add an increasing number of counters:
- From Pattern 1 to Pattern 2, the increment is 3 (6 - 3)
- From Pattern 2 to Pattern 3, the increment is 4 (10 - 6)
- From Pattern 3 to Pattern 4, the increment is 5 (15 - 10)
This suggests that the increments are increasing by 1 each time. Therefore, we can predict:
- Pattern 5 (15 + 6 = 21 counters)
- Pattern 6 (21 + 7 = 28 counters)
In total, Pattern 5 and Pattern 6 require:
Since Ryan started with 80 counters, he still has enough counters. Therefore, Ryan is correct because 80 is greater than 49.
Step 2
Complete the table for the addition of counters in consecutive patterns
Answer
Using the provided patterns:
-
For Pattern 1 + Pattern 2:
- Counters to add: 3 + 6 = 9
- Total counters: 9
-
For Pattern 2 + Pattern 3:
- Counters to add: 6 + 10 = 16
- Total counters: 16
-
For Pattern 3 + Pattern 4:
- Counters to add: 10 + 15 = 25
- Total counters: 25
Thus, the completed table is:
| Patterns to add | Counters to add | Total counters |
|---|---|---|
| Pattern 1 + Pattern 2 | 3 + 6 | 9 |
| Pattern 2 + Pattern 3 | 6 + 10 | 16 |
| Pattern 3 + Pattern 4 | 10 + 15 | 25 |
Step 3
Find the value of k.
Answer
Given the equation:
Number of counters in Pattern k + Pattern (k + 1) = 144,
We know that the number of counters follows the pattern of adding incrementally:
- The formula for the number of counters for Pattern k can be derived from the previous analysis: the sum is:
n = rac{(n)(n+1)}{2} for triangle numbers; however, we will use the pattern derived from previous calculations.
Assuming Pattern k has some linear relation, we express it:
For k=1:
- Pattern 1 + Pattern 2 = 3 + 6 = 9
Continuing this, we can solve for when:
- Setting up the equation: Pattern k + Pattern (k + 1) = 3k + 3(k + 1) = 144.
Simplifying gives us:
Solving for k, we get:
This doesn't yield an integer, thus iterating or further deduction may be needed.
However, if we had started counting from zero, running through the values one more could lead to detecting k more readily. Therefore, solving would involve checking integer values to see when: Should k turn back into a pattern count, round down to discover the pattern in sequence. The pivotal k should round to the nearest integer that satisfies this, which leads us to: .