Math worksheet focusing on number sequences with various exercises and a playful illustration of mathematical symbols.

Let's solve each question step by step with clear explanations.

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1) Write down the next two terms in the sequence: 3, 15, 75, ____, ____

Step-by-step analysis:

Look at the pattern:

- 3 → 15: $ 3 \times 5 = 15 $
- 15 → 75: $ 15 \times 5 = 75 $

So, it's multiplying by 5 each time.

- Next term: $ 75 \times 5 = 375 $
- Next term: $ 375 \times 5 = 1875 $

✔ Answer: 375, 1875

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2) Find the next two terms in the sequence: 9, 8, 10, 9, 11, ____, ____

Step-by-step analysis:

List the terms with positions:

| Position | Term |
|----------|------|
| 1 | 9 |
| 2 | 8 |
| 3 | 10 |
| 4 | 9 |
| 5 | 11 |

Now observe:

- Odd positions (1st, 3rd, 5th): 9, 10, 11 → increasing by 1
- Even positions (2nd, 4th): 8, 9 → increasing by 1

So:
- 6th term (even): next after 9 → 10
- 7th term (odd): next after 11 → 12

✔ Answer: 10, 12

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3) Find the next two terms in the sequence: 1, 8, 27, 64, ____, ____

Step-by-step analysis:

Recognize these numbers:

- $ 1 = 1^3 $
- $ 8 = 2^3 $
- $ 27 = 3^3 $
- $ 64 = 4^3 $

So this is the cubes of natural numbers.

Next:
- $ 5^3 = 125 $
- $ 6^3 = 216 $

✔ Answer: 125, 216

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4) Determine the term-to-term rule for the following sequence: -6, -4, -2, 0, 2, ...

Step-by-step analysis:

Look at the differences:

- $ -4 - (-6) = 2 $
- $ -2 - (-4) = 2 $
- $ 0 - (-2) = 2 $
- $ 2 - 0 = 2 $

Each term increases by +2.

So the rule is: Add 2 to get the next term.

✔ Rule: Add 2

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5) Determine the term-to-term rule for the sequence: -8, -4, -2, -1, -½, ...

Step-by-step analysis:

Check ratios or patterns:

- $ -8 $ to $ -4 $: $ -4 / -8 = 0.5 $ → multiplied by 0.5
- $ -4 $ to $ -2 $: $ -2 / -4 = 0.5 $
- $ -2 $ to $ -1 $: $ -1 / -2 = 0.5 $
- $ -1 $ to $ -\frac{1}{2} $: $ (-0.5)/(-1) = 0.5 $

So each term is multiplied by ½.

✔ Rule: Multiply by $ \frac{1}{2} $

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6) Use the following nth terms to find the first four sequences and the tenth term (10th):

We plug in $ n = 1, 2, 3, 4 $ for first four terms, and $ n = 10 $ for 10th term.

#### a) $ 3n + 1 $

- $ n=1 $: $ 3(1)+1 = 4 $
- $ n=2 $: $ 3(2)+1 = 7 $
- $ n=3 $: $ 3(3)+1 = 10 $
- $ n=4 $: $ 3(4)+1 = 13 $
- $ n=10 $: $ 3(10)+1 = 31 $

✔ First four: 4, 7, 10, 13; 10th term: 31

#### b) $ 2n - 1 $

- $ n=1 $: $ 2(1)-1 = 1 $
- $ n=2 $: $ 2(2)-1 = 3 $
- $ n=3 $: $ 2(3)-1 = 5 $
- $ n=4 $: $ 2(4)-1 = 7 $
- $ n=10 $: $ 2(10)-1 = 19 $

✔ First four: 1, 3, 5, 7; 10th term: 19

#### c) $ -2n + 30 $

- $ n=1 $: $ -2(1)+30 = 28 $
- $ n=2 $: $ -2(2)+30 = 26 $
- $ n=3 $: $ -2(3)+30 = 24 $
- $ n=4 $: $ -2(4)+30 = 22 $
- $ n=10 $: $ -2(10)+30 = 10 $

✔ First four: 28, 26, 24, 22; 10th term: 10

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7) Find the nth term for the following sequences:

#### a) 3, 7, 11, 15, 19

Step-by-step:

- Common difference: $ 7 - 3 = 4 $
- So it’s an arithmetic sequence with $ d = 4 $

General form: $ a_n = a + (n-1)d $

- $ a = 3 $, $ d = 4 $
- $ a_n = 3 + (n-1)(4) = 3 + 4n - 4 = 4n - 1 $

✔ nth term: $ 4n - 1 $

#### b) 13, 15, 17, 19

- Common difference: $ 15 - 13 = 2 $
- $ a = 13 $, $ d = 2 $
- $ a_n = 13 + (n-1)(2) = 13 + 2n - 2 = 2n + 11 $

✔ nth term: $ 2n + 11 $

#### c) 20, 16, 12, 8

- Common difference: $ 16 - 20 = -4 $
- $ a = 20 $, $ d = -4 $
- $ a_n = 20 + (n-1)(-4) = 20 - 4n + 4 = -4n + 24 $

✔ nth term: $ -4n + 24 $

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✔ Final Answers Summary:

1) 375, 1875
2) 10, 12
3) 125, 216
4) Add 2
5) Multiply by ½
6)
a) 4, 7, 10, 13; 10th: 31
b) 1, 3, 5, 7; 10th: 19
c) 28, 26, 24, 22; 10th: 10
7)
a) $ 4n - 1 $
b) $ 2n + 11 $
c) $ -4n + 24 $

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