jmaths
MYP 5 Extended Mathematics · Unit 1 Sequences · 10.1.1 Patterns and generalising
Name Date
Patterns in shapes and in numbers
The same patterns three ways: first as shapes, then as numbers, then numbers on their own.
Side 1 Shapes
For each pattern: write how many pieces are in Patterns 1 to 5, then Pattern 10, then Pattern \(n\). Say how you see it in the shape. One pattern has a term-to-term rule only: continue it, do not look for a rule for \(n\).
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A
1 2 3 4 5 10 \(n\) How I see it:
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B
1 2 3 4 5 10 \(n\) How I see it:
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C
1 2 3 4 5 10 \(n\) How I see it:
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D
1 2 3 4 5 10 \(n\) How I see it:
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E
1 2 3 4 5 10 \(n\) term-to-term How I see it:
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F
1 2 3 4 5 10 \(n\) How I see it:
Side 2 The same patterns as numbers
These are the six sequences from side 1, in a different order and with no shapes. Complete each row and write the rule. Then say which shape it was. Shaded boxes are not needed.
| Sequence | \(n = 1\) | \(n = 2\) | \(n = 3\) | \(n = 4\) | \(n = 5\) | \(n = 10\) | \(n = 100\) | \(n\)th term | Shape |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 8 | 27 | 64 | 125 | ||||
| 2 | 4 | 8 | 12 | 16 | 20 | ||||
| 3 | 1 | 3 | 6 | 10 | 15 | term-to-term | |||
| 4 | 2 | 6 | 12 | 20 | 30 | ||||
| 5 | 4 | 9 | 16 | 25 | 36 | ||||
| 6 | 5 | 12 | 21 | 32 | 45 |
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Which sequence was easier to find a rule for from the shape than from the numbers? Which was easier from the numbers?
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Choose one sequence. Use its shape to explain why its rule must work for every pattern, not only the ones drawn.
You need both. The numbers show what the rule is. The shape shows why it works.
Side 3 Numbers on their own
No shapes this time. Look for how each sequence is built, then use what you spot.
Part A General rules: complete each row, then write the rule for the \(n\)th term
| Sequence | \(n = 1\) | \(n = 2\) | \(n = 3\) | \(n = 4\) | \(n = 5\) | \(n = 10\) | \(n = 100\) | \(n\)th term |
|---|---|---|---|---|---|---|---|---|
| a | 4 | 8 | 12 | 16 | 20 | |||
| b | 1 | 4 | 9 | 16 | 25 | |||
| c | 2 | 6 | 12 | 20 | 30 | |||
| d | 2 | 5 | 10 | 17 | 26 | |||
| e | 0 | 4 | 8 | 12 | 16 | |||
| f | 1 | 8 | 27 | 64 | 125 | |||
| g | 4 | 9 | 16 | 25 | 36 | |||
| h | 5 | 12 | 21 | 32 | 45 | |||
| i | 0 | 7 | 26 | 63 | 124 | |||
| j | 2 | 8 | 18 | 32 | 50 |
Part B Term-to-term rules: write the next three terms, then say how each term comes from the ones before
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a
1,3,6,10,15,
How it grows:
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b
1,1,2,3,5,8,
How it grows:
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c
1,2,4,7,11,16,
How it grows:
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d
2,1,3,4,7,11,
How it grows:
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e
1,4,10,20,35,
How it grows:
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f
3,4,7,11,18,
How it grows:
Part C Verify and justify
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Ali says the rule for sequence h in Part A is \(n^2 + 4n\). Bea says it is \((n+2)^2 - 4\). Test both rules with \(n = 6\) and \(n = 7\). Who is right?
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Compare sequence c in Part A with sequence a in Part B. What do you notice? Check it for the first five terms.
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Sequence i in Part A is always 1 less than sequence f. Explain why, using their rules.
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Is 400 a term of sequence g in Part A? Is 200 a term of sequence b? Say how you know.