Patterns, 5th grade - worksheets - Individualized Math by Destiny ... - Free Printable

Since I can't view images directly, I’ll solve the problems based on the text you provided. Here's the full solution for the Mastery Check: Patterns worksheet.

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Part 1: Continue each pattern and write whether it is arithmetic, geometric, or neither

We will analyze each sequence, find the pattern, continue it, and classify it.

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#### 1. 24, 34, 33, 43, 42, ____, ____, ____, ____, ____

Let’s look at the pattern:

- 24 → 34 (+10)
- 34 → 33 (−1)
- 33 → 43 (+10)
- 43 → 42 (−1)

So the pattern alternates: +10, −1, +10, −1, ...

Continue:
- 42 → 52 (+10)
- 52 → 51 (−1)
- 51 → 61 (+10)
- 61 → 60 (−1)
- 60 → 70 (+10)

Sequence: 24, 34, 33, 43, 42, 52, 51, 61, 60, 70

Type: Neither (alternating operations, not constant difference or ratio)

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#### 2. 1, 1, 2, 3, 5, ____, ____, ____, ____, ____

This is the Fibonacci sequence:
Each term is the sum of the two previous terms.

- 1 + 1 = 2
- 1 + 2 = 3
- 2 + 3 = 5
- 3 + 5 = 8
- 5 + 8 = 13
- 8 + 13 = 21
- 13 + 21 = 34
- 21 + 34 = 55

Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

Type: Neither (each term depends on two prior terms — not arithmetic or geometric)

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#### 3. 76, 69, 62, 55, 48, ____, ____, ____, ____, ____

Check differences:
- 76 → 69 (−7)
- 69 → 62 (−7)
- 62 → 55 (−7)
- 55 → 48 (−7)

Common difference: −7 → Arithmetic sequence

Continue:
- 48 − 7 = 41
- 41 − 7 = 34
- 34 − 7 = 27
- 27 − 7 = 20
- 20 − 7 = 13

Sequence: 76, 69, 62, 55, 48, 41, 34, 27, 20, 13

Type: Arithmetic

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#### 4. 7, 21, 63, 189, 567, ____, ____, ____, ____, ____

Check ratios:
- 21 ÷ 7 = 3
- 63 ÷ 21 = 3
- 189 ÷ 63 = 3
- 567 ÷ 189 = 3

Common ratio: ×3 → Geometric sequence

Continue:
- 567 × 3 = 1701
- 1701 × 3 = 5103
- 5103 × 3 = 15309
- 15309 × 3 = 45927
- 45927 × 3 = 137781

Sequence: 7, 21, 63, 189, 567, 1701, 5103, 15309, 45927, 137781

Type: Geometric

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#### 5. 5, 10, 20, 35, 55, ____, ____, ____, ____, ____

Check differences:
- 10 − 5 = 5
- 20 − 10 = 10
- 35 − 20 = 15
- 55 − 35 = 20

Differences: 5, 10, 15, 20 → increasing by 5 each time

Next difference: 25 → 55 + 25 = 80
Then: 30 → 80 + 30 = 110
Then: 35 → 110 + 35 = 145
Then: 40 → 145 + 40 = 185
Then: 45 → 185 + 45 = 230

Sequence: 5, 10, 20, 35, 55, 80, 110, 145, 185, 230

Type: Neither (second differences are constant → quadratic-like, but not arithmetic or geometric)

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#### 6. 70, 69, 67, 64, 60, ____, ____, ____, ____, ____

Check differences:
- 70 → 69 (−1)
- 69 → 67 (−2)
- 67 → 64 (−3)
- 64 → 60 (−4)

Pattern: subtracting 1, 2, 3, 4, ... so next subtractions: −5, −6, −7, −8, −9

- 60 − 5 = 55
- 55 − 6 = 49
- 49 − 7 = 42
- 42 − 8 = 34
- 34 − 9 = 25

Sequence: 70, 69, 67, 64, 60, 55, 49, 42, 34, 25

Type: Neither (decreasing by increasing amounts)

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#### 7. 1, 4, 9, 16, 25, ____, ____, ____, ____, ____

These are perfect squares:
- 1² = 1
- 2² = 4
- 3² = 9
- 4² = 16
- 5² = 25

So next: 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100

Sequence: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Type: Neither (squares — not linear or multiplicative)

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Part 2: Write the next five terms of each pattern, given the rule and starting number(s)

> 8. Rule: ×2, −6
> Starting number: 6

Apply the rule step-by-step:

- Start: 6
- ×2 = 12, then −6 = 6
- ×2 = 12, then −6 = 6
- Repeat...

Wait — let’s do this carefully:

Rule: Apply ×2, then −6 to each term.

Start: 6

1. 6 → (6 × 2) − 6 = 12 − 6 = 6
2. 6 → (6 × 2) − 6 = 12 − 6 = 6
3. 6 → same → 6
4. → 6
5. → 6

So the sequence becomes: 6, 6, 6, 6, 6, ...

But that seems trivial. Let’s double-check the rule.

Possibility: The rule is applied sequentially: each term is obtained by taking the previous term, multiplying by 2, then subtracting 6.

So:

- Term 1: 6
- Term 2: (6 × 2) − 6 = 12 − 6 = 6
- Term 3: (6 × 2) − 6 = 6
- etc.

So yes, it stabilizes at 6.

But maybe we misread? Let’s suppose the rule is meant to be a two-step operation applied once per term.

Alternatively, perhaps the rule is: multiply by 2, then subtract 6, and apply to each term.

Yes, as above.

So:

Next five terms: 6, 6, 6, 6, 6

But this feels odd. Maybe there’s a typo?

Wait — perhaps the rule is meant to be "multiply by 2, then subtract 6", and we start from 6.

But as shown, it stays at 6.

Alternatively, maybe the rule is applied in order, like a function: f(n) = 2n − 6

Then:

- Start: 6
- Next: 2(6) − 6 = 12 − 6 = 6
- Then: 2(6) − 6 = 6 → stuck

So yes, it's a fixed point.

So the sequence is constant: 6, 6, 6, 6, 6

But let's confirm with another example.

Suppose start was 7:

- 2×7 − 6 = 14 − 6 = 8
- 2×8 − 6 = 16 − 6 = 10
- 2×10 − 6 = 20 − 6 = 14
- 2×14 − 6 = 28 − 6 = 22
- 2×22 − 6 = 44 − 6 = 38

But here, start is 6 → always 6.

So answer:

Next five terms: 6, 6, 6, 6, 6

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✔ Final Answers

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Part 1: Complete Sequences & Type

| # | Sequence | Type |
|---|--------|------|
| 1 | 24, 34, 33, 43, 42, 52, 51, 61, 60, 70 | Neither |
| 2 | 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 | Neither |
| 3 | 76, 69, 62, 55, 48, 41, 34, 27, 20, 13 | Arithmetic |
| 4 | 7, 21, 63, 189, 567, 1701, 5103, 15309, 45927, 137781 | Geometric |
| 5 | 5, 10, 20, 35, 55, 80, 110, 145, 185, 230 | Neither |
| 6 | 70, 69, 67, 64, 60, 55, 49, 42, 34, 25 | Neither |
| 7 | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 | Neither |

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Part 2: Rule-based Pattern

8. Rule: ×2, −6; Start: 6

Apply:
- 6 → (6 × 2) − 6 = 6
- 6 → (6 × 2) − 6 = 6
- 6 → 6
- 6 → 6
- 6 → 6

Next five terms: 6, 6, 6, 6, 6

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Let me know if you'd like a visual explanation or help with any specific part!