High School Math Worksheets | Math Worksheets PDF - Free Printable

Let's solve each section of the worksheet "Generating Sequences" step by step. The goal is to generate terms of sequences using given formulas by substituting values of $ n = 1, 2, 3, \ldots $ as required.

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🔷 Section A: Find the first five terms

We substitute $ n = 1, 2, 3, 4, 5 $ into each expression.

| Expression | $ n=1 $ | $ n=2 $ | $ n=3 $ | $ n=4 $ | $ n=5 $ |
|----------|--------|--------|--------|--------|--------|
| $ n + 2 $ | $ 1+2 = 3 $ | $ 2+2 = 4 $ | $ 3+2 = 5 $ | $ 4+2 = 6 $ | $ 5+2 = 7 $ |
| $ n - 7 $ | $ 1-7 = -6 $ | $ 2-7 = -5 $ | $ 3-7 = -4 $ | $ 4-7 = -3 $ | $ 5-7 = -2 $ |
| $ 4n $ | $ 4(1) = 4 $ | $ 4(2) = 8 $ | $ 4(3) = 12 $ | $ 4(4) = 16 $ | $ 4(5) = 20 $ |
| $ 3n + 1 $ | $ 3(1)+1 = 4 $ | $ 3(2)+1 = 7 $ | $ 3(3)+1 = 10 $ | $ 3(4)+1 = 13 $ | $ 3(5)+1 = 16 $ |
| $ 5n - 8 $ | $ 5(1)-8 = -3 $ | $ 5(2)-8 = 2 $ | $ 5(3)-8 = 7 $ | $ 5(4)-8 = 12 $ | $ 5(5)-8 = 17 $ |
| $ 2n + 0.5 $ | $ 2(1)+0.5 = 2.5 $ | $ 2(2)+0.5 = 4.5 $ | $ 2(3)+0.5 = 6.5 $ | $ 2(4)+0.5 = 8.5 $ | $ 2(5)+0.5 = 10.5 $ |
| $ 12n - 1.5 $ | $ 12(1)-1.5 = 10.5 $ | $ 12(2)-1.5 = 22.5 $ | $ 12(3)-1.5 = 34.5 $ | $ 12(4)-1.5 = 46.5 $ | $ 12(5)-1.5 = 58.5 $ |

✔ Section A Completed

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🔷 Section B: Find the first five terms

Same process.

| Expression | $ n=1 $ | $ n=2 $ | $ n=3 $ | $ n=4 $ | $ n=5 $ |
|----------|--------|--------|--------|--------|--------|
| $ 2 - n $ | $ 2-1 = 1 $ | $ 2-2 = 0 $ | $ 2-3 = -1 $ | $ 2-4 = -2 $ | $ 2-5 = -3 $ |
| $ 13 - n $ | $ 13-1 = 12 $ | $ 13-2 = 11 $ | $ 13-3 = 10 $ | $ 13-4 = 9 $ | $ 13-5 = 8 $ |
| $ -3n $ | $ -3(1) = -3 $ | $ -3(2) = -6 $ | $ -3(3) = -9 $ | $ -3(4) = -12 $ | $ -3(5) = -15 $ |
| $ 1 - 2n $ | $ 1-2(1) = -1 $ | $ 1-2(2) = -3 $ | $ 1-2(3) = -5 $ | $ 1-2(4) = -7 $ | $ 1-2(5) = -9 $ |
| $ -5n + 3 $ | $ -5(1)+3 = -2 $ | $ -5(2)+3 = -7 $ | $ -5(3)+3 = -12 $ | $ -5(4)+3 = -17 $ | $ -5(5)+3 = -22 $ |
| $ \frac{1}{2}n $ | $ 0.5(1) = 0.5 $ | $ 0.5(2) = 1 $ | $ 0.5(3) = 1.5 $ | $ 0.5(4) = 2 $ | $ 0.5(5) = 2.5 $ |
| $ \frac{3}{4}n $ | $ 0.75(1) = 0.75 $ | $ 0.75(2) = 1.5 $ | $ 0.75(3) = 2.25 $ | $ 0.75(4) = 3 $ | $ 0.75(5) = 3.75 $ |

✔ Section B Completed

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🔷 Section C: First four terms

Now for $ n = 1, 2, 3, 4 $

1) $ 2n - 12 $
→ $ 2(1)-12 = -10 $, $ 2(2)-12 = -8 $, $ 2(3)-12 = -6 $, $ 2(4)-12 = -4 $
→ -10, -8, -6, -4

2) $ 6n - 10 $
→ $ 6(1)-10 = -4 $, $ 6(2)-10 = 2 $, $ 6(3)-10 = 8 $, $ 6(4)-10 = 14 $
→ -4, 2, 8, 14

3) $ n - 98 $
→ $ 1-98 = -97 $, $ 2-98 = -96 $, $ 3-98 = -95 $, $ 4-98 = -94 $
→ -97, -96, -95, -94

4) $ 3n + 7 $
→ $ 3(1)+7 = 10 $, $ 3(2)+7 = 13 $, $ 3(3)+7 = 16 $, $ 3(4)+7 = 19 $
→ 10, 13, 16, 19

5) $ 2.5n $
→ $ 2.5(1) = 2.5 $, $ 2.5(2) = 5 $, $ 2.5(3) = 7.5 $, $ 2.5(4) = 10 $
→ 2.5, 5, 7.5, 10

6) $ 4 - 2n $
→ $ 4-2(1) = 2 $, $ 4-2(2) = 0 $, $ 4-2(3) = -2 $, $ 4-2(4) = -4 $
→ 2, 0, -2, -4

7) $ 8 - n $
→ $ 8-1 = 7 $, $ 8-2 = 6 $, $ 8-3 = 5 $, $ 8-4 = 4 $
→ 7, 6, 5, 4

8) $ 0.5n + 3 $
→ $ 0.5(1)+3 = 3.5 $, $ 0.5(2)+3 = 4 $, $ 0.5(3)+3 = 4.5 $, $ 0.5(4)+3 = 5 $
→ 3.5, 4, 4.5, 5

9) $ 6 - 3n $
→ $ 6-3(1) = 3 $, $ 6-3(2) = 0 $, $ 6-3(3) = -3 $, $ 6-3(4) = -6 $
→ 3, 0, -3, -6

10) $ 6n $
→ $ 6(1)=6 $, $ 6(2)=12 $, $ 6(3)=18 $, $ 6(4)=24 $
→ 6, 12, 18, 24

11) $ \frac{n}{2} $
→ $ \frac{1}{2} = 0.5 $, $ \frac{2}{2} = 1 $, $ \frac{3}{2} = 1.5 $, $ \frac{4}{2} = 2 $
→ 0.5, 1, 1.5, 2

12) $ 8 - 5n $
→ $ 8-5(1) = 3 $, $ 8-5(2) = -2 $, $ 8-5(3) = -7 $, $ 8-5(4) = -12 $
→ 3, -2, -7, -12

✔ Section C Completed

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🔷 Section D (Level 7!): First three terms

Now for $ n = 1, 2, 3 $

1) $ 2(n + 1) $
→ $ 2(1+1) = 4 $, $ 2(2+1) = 6 $, $ 2(3+1) = 8 $ → 4, 6, 8

2) $ n^2 $
→ $ 1^2 = 1 $, $ 2^2 = 4 $, $ 3^2 = 9 $ → 1, 4, 9

3) $ \frac{n+1}{n} $
→ $ \frac{2}{1} = 2 $, $ \frac{3}{2} = 1.5 $, $ \frac{4}{3} \approx 1.333 $ → 2, 1.5, 1.\overline{3}

4) $ 3(n + 5) $
→ $ 3(6) = 18 $, $ 3(7) = 21 $, $ 3(8) = 24 $ → 18, 21, 24

5) $ n^2 + 1 $
→ $ 1+1 = 2 $, $ 4+1 = 5 $, $ 9+1 = 10 $ → 2, 5, 10

6) $ \frac{3n}{2} $
→ $ \frac{3}{2} = 1.5 $, $ \frac{6}{2} = 3 $, $ \frac{9}{2} = 4.5 $ → 1.5, 3, 4.5

7) $ 10(n - 9) $
→ $ 10(-8) = -80 $, $ 10(-7) = -70 $, $ 10(-6) = -60 $ → -80, -70, -60

8) $ n^2 + 5 $
→ $ 1+5 = 6 $, $ 4+5 = 9 $, $ 9+5 = 14 $ → 6, 9, 14

9) $ n(n + 2) $
→ $ 1(3) = 3 $, $ 2(4) = 8 $, $ 3(5) = 15 $ → 3, 8, 15

10) $ 25(1 - n) $
→ $ 25(0) = 0 $, $ 25(-1) = -25 $, $ 25(-2) = -50 $ → 0, -25, -50

11) $ 2n^2 $
→ $ 2(1) = 2 $, $ 2(4) = 8 $, $ 2(9) = 18 $ → 2, 8, 18

12) $ n^3 $
→ $ 1^3 = 1 $, $ 2^3 = 8 $, $ 3^3 = 27 $ → 1, 8, 27

✔ Section D Completed

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✔ Final Answers Summary

#### Section A
| Expression | Terms |
|---------|-------|
| $ n+2 $ | 3, 4, 5, 6, 7 |
| $ n-7 $ | -6, -5, -4, -3, -2 |
| $ 4n $ | 4, 8, 12, 16, 20 |
| $ 3n+1 $ | 4, 7, 10, 13, 16 |
| $ 5n-8 $ | -3, 2, 7, 12, 17 |
| $ 2n+0.5 $ | 2.5, 4.5, 6.5, 8.5, 10.5 |
| $ 12n-1.5 $ | 10.5, 22.5, 34.5, 46.5, 58.5 |

#### Section B
| Expression | Terms |
|---------|-------|
| $ 2-n $ | 1, 0, -1, -2, -3 |
| $ 13-n $ | 12, 11, 10, 9, 8 |
| $ -3n $ | -3, -6, -9, -12, -15 |
| $ 1-2n $ | -1, -3, -5, -7, -9 |
| $ -5n+3 $ | -2, -7, -12, -17, -22 |
| $ \frac{1}{2}n $ | 0.5, 1, 1.5, 2, 2.5 |
| $ \frac{3}{4}n $ | 0.75, 1.5, 2.25, 3, 3.75 |

#### Section C
1. $ 2n-12 $: -10, -8, -6, -4
2. $ 6n-10 $: -4, 2, 8, 14
3. $ n-98 $: -97, -96, -95, -94
4. $ 3n+7 $: 10, 13, 16, 19
5. $ 2.5n $: 2.5, 5, 7.5, 10
6. $ 4-2n $: 2, 0, -2, -4
7. $ 8-n $: 7, 6, 5, 4
8. $ 0.5n+3 $: 3.5, 4, 4.5, 5
9. $ 6-3n $: 3, 0, -3, -6
10. $ 6n $: 6, 12, 18, 24
11. $ \frac{n}{2} $: 0.5, 1, 1.5, 2
12. $ 8-5n $: 3, -2, -7, -12

#### Section D
1. $ 2(n+1) $: 4, 6, 8
2. $ n^2 $: 1, 4, 9
3. $ \frac{n+1}{n} $: 2, 1.5, 1.333...
4. $ 3(n+5) $: 18, 21, 24
5. $ n^2+1 $: 2, 5, 10
6. $ \frac{3n}{2} $: 1.5, 3, 4.5
7. $ 10(n-9) $: -80, -70, -60
8. $ n^2+5 $: 6, 9, 14
9. $ n(n+2) $: 3, 8, 15
10. $ 25(1-n) $: 0, -25, -50
11. $ 2n^2 $: 2, 8, 18
12. $ n^3 $: 1, 8, 27

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📝 Explanation:

To generate a sequence from a formula:
1. Plug in successive values of $ n $ starting from 1.
2. Simplify the expression for each $ n $.
3. Write down the results as terms of the sequence.

This method works for linear, quadratic, rational, and higher-degree expressions. For fractions or decimals, ensure accuracy in computation.

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