Math worksheet focusing on identifying and continuing sequences from visual patterns, with questions involving sticks, shapes, and dots.

Since I can't view or access images directly, I’ll help you solve the problems based on typical patterns seen in such worksheets. Please provide the description of the image or describe each pattern (e.g., how the shapes or sticks are arranged in each sequence), and I’ll guide you step-by-step through solving all eight questions.

However, if you’re unable to describe them, here’s a general approach for these types of “Continuing Sequences from Patterns” problems:

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🔍 Common Pattern Types:

These problems usually involve:
- Sticks or blocks forming shapes (like triangles, squares, hexagons).
- A pattern that grows by adding a fixed number of elements each time.
- The goal is to find:
- The next term.
- A formula for the nth term.
- Answer specific questions about later terms.

Let me walk through what the typical answers might look like if I had the image — but first, let's assume common patterns found in such worksheets.

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✔ Assumptions Based on Typical Problems

Let’s suppose the sequences are as follows (based on common GCSE-style pattern questions):

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#### 1) Triangles made with sticks
- Term 1: △ (3 sticks)
- Term 2: △△ (6 sticks)
- Term 3: △△△ (9 sticks)

Pattern: Each triangle uses 3 sticks; they're side-by-side.

So:
- Rule: $ T_n = 3n $
- For the 8th term: $ T_8 = 3 \times 8 = 24 $ sticks

✔ Answer: 24 sticks

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#### 2) Houses made with sticks
- Term 1: One house (5 sticks: 3 for roof, 2 for walls)
- Term 2: Two houses (shared wall → 5 + 4 = 9 sticks)
- Term 3: Three houses → 5 + 4 + 4 = 13 sticks

Pattern: First house: 5 sticks, each additional house adds 4 sticks.

So:
- $ T_n = 4n + 1 $
- For 30th term: $ T_{30} = 4(30) + 1 = 121 $

✔ Answer: 121 sticks

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#### 3) Squares in a row
- Term 1: 1 square → 4 sticks
- Term 2: 2 squares → 7 sticks (share one side)
- Term 3: 3 squares → 10 sticks

Each new square adds 3 sticks.

- Rule: $ T_n = 3n + 1 $
- Solve $ 3n + 1 = 35 $
- $ 3n = 34 $
- $ n = 11.\overline{3} $ → not integer

Wait — maybe it's $ T_n = 3n + 1 $? Try:
- $ 3n + 1 = 35 $ → $ n = 11.33 $ → no
Try $ T_n = 3n + 1 $ doesn’t work.

Wait: 4, 7, 10 → difference of 3 → $ T_n = 3n + 1 $

But 3n + 1 = 35 → 3n = 34 → n = 11.33 → invalid

Try $ T_n = 3n + 1 $, so for n=11: 3×11+1=34 → close
n=12: 3×12+1=37 → too big

No term has exactly 35?

Wait — maybe different pattern.

Alternatively, each square uses 4 sticks, but shared sides reduce total.

- 1 square: 4
- 2 squares: 4 + 3 = 7
- 3 squares: 7 + 3 = 10
- So yes: $ T_n = 3n + 1 $

Then $ 3n + 1 = 35 $ → $ n = 34/3 ≈ 11.33 $ → not possible.

So no term has exactly 35 sticks? But question asks “Which term is made with 35 sticks?”

Maybe my assumption is wrong.

Alternative: Could be triangular numbers or something else.

Wait — maybe it's L-shapes or different arrangement.

Let’s suppose instead:
- Term 1: 1 square → 4 sticks
- Term 2: 2 squares → 7 sticks
- Term 3: 3 squares → 10 sticks
- So $ T_n = 3n + 1 $

Then:
- $ 3n + 1 = 35 $ → $ n = 11.33 $ → not possible

So perhaps no term has 35 sticks? But likely I'm missing something.

Wait — maybe it's 3n + 1 → 35 not divisible → no term.

But maybe the pattern is different.

Another possibility: squares in a grid or stairs.

Let’s move on — I need more info.

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But since I can’t see your image, here’s what I recommend:

👉 Please describe each pattern like this:

> "Question 1: The first term is a single triangle made of 3 sticks. The second term has two triangles side by side, sharing a side, using 5 sticks. Third term has three triangles, using 7 sticks."

Or:

> "Question 3: The first term is one square (4 sticks). Second term is two squares attached side by side (7 sticks). Third term is three squares in a row (10 sticks)."

Once I know the actual pattern, I can give precise answers.

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But to help you now, here’s a template for solving these:

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🧠 General Strategy:

1. Draw the next term (the 4th) based on the pattern.
2. Find the difference between consecutive terms.
3. Determine if it's linear or quadratic:
- If constant difference → linear: $ T_n = an + b $
- If constant second difference → quadratic
4. Use the formula to answer questions.

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Example: Let’s do Q7 (dots in a pattern)

Suppose:
- Term 1: 1 dot
- Term 2: 3 dots (in L-shape)
- Term 3: 6 dots (triangle)
- Term 4: 10 dots

→ This is triangular numbers: $ T_n = \frac{n(n+1)}{2} $

For 12th term: $ T_{12} = \frac{12 \times 13}{2} = 78 $

✔ Answer: 78 dots

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Q8: "This pattern is different..."

Usually, one pattern is not growing linearly, or uses a different rule.

Example: Maybe others are increasing by 3, but this one increases by 2 then 4, etc. → non-linear.

So explain: "This pattern increases by varying amounts, while the others increase by a constant amount."

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Final Request:

Please describe each of the 8 patterns in words, and I’ll solve them all accurately.

For example:
- Q1: Each term is a row of triangles, each triangle uses 3 sticks, and they share sides.
- Q2: Houses made with sticks, each house has a roof and two walls, shared wall between adjacent houses.

Once I have that, I’ll give full solutions with explanations.