The image you've shared is an educational slide explaining
sequences in mathematics, with examples of different types of sequences. Let's break down the content and solve any implied task or understanding exercise based on it.
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🔍 Understanding the Slide
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✔ Definition:
> A
sequence is a set of numbers or algebraic terms that follow a particular pattern or rule to get from one term to the next.
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📌 Examples Given:
#### 1.
Arithmetic Sequence
Sequence:
2, 5, 8, 11, 14, ...
- Pattern: Each term increases by
+3.
- Rule: Add 3 to get the next term.
- This is called an
arithmetic sequence because the difference between consecutive terms is constant.
✔ Next term: 14 + 3 =
17
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#### 2.
Geometric Sequence
Sequence:
3, 6, 12, 24, 48
- Pattern: Each term is multiplied by
×2.
- Rule: Multiply by 2 to get the next term.
- This is a
geometric sequence because the ratio between consecutive terms is constant.
✔ Next term: 48 × 2 =
96
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#### 3.
Triangular Numbers
Sequence:
1, 3, 6, 10, ...
- Represented visually as dots forming triangles.
- Pattern: The differences between terms increase by 1 each time:
- 1 → 3: +2
- 3 → 6: +3
- 6 → 10: +4
- So the next difference would be
+5, giving:
- 10 + 5 =
15
✔ Next term:
15
This sequence is known as
triangular numbers, where the $n$th term is given by the formula:
$$
T_n = \frac{n(n+1)}{2}
$$
For example:
- $T_1 = \frac{1(1+1)}{2} = 1$
- $T_2 = \frac{2(2+1)}{2} = 3$
- $T_3 = \frac{3(3+1)}{2} = 6$
- $T_4 = \frac{4(4+1)}{2} = 10$
- $T_5 = \frac{5(5+1)}{2} = 15$
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✔ Summary of Key Concepts:
| Type | Rule | Example | Next Term |
|------------------|------------------------------|--------------------------------|-----------|
| Arithmetic | Add a constant number | 2, 5, 8, 11, 14, ... | 17 |
| Geometric | Multiply by a constant factor| 3, 6, 12, 24, 48, ... | 96 |
| Triangular | Add increasing integers | 1, 3, 6, 10, ... | 15 |
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🧠 Task (Implied): Identify the type of sequence and find the next term.
If the task was to
complete the sequences, here are the answers:
1.
2, 5, 8, 11, 14, ... → Next term:
17 (arithmetic)
2.
3, 6, 12, 24, 48, ... → Next term:
96 (geometric)
3.
1, 3, 6, 10, ... → Next term:
15 (triangular)
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💡 Final Answer:
> The three sequences shown are:
> -
Arithmetic: add 3 → next term is
17
> -
Geometric: multiply by 2 → next term is
96
> -
Triangular numbers: add increasing numbers (+2, +3, +4, +5...) → next term is
15
Let me know if you'd like to generate a formula for any of these sequences or explore more patterns!
Parent Tip: Review the logic above to help your child master the concept of sequences worksheet gcse.
