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Problem: Solving the Sequences Worksheet

The task involves generating the first few terms of various sequences based on given formulas. Let's solve each section step by step.

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#### Section A: Find the first five terms of the following sequences.

We are given formulas and need to compute the values for \( n = 1, 2, 3, 4, 5 \).

1. \( n + 2 \)
- For \( n = 1 \): \( 1 + 2 = 3 \)
- For \( n = 2 \): \( 2 + 2 = 4 \)
- For \( n = 3 \): \( 3 + 2 = 5 \)
- For \( n = 4 \): \( 4 + 2 = 6 \)
- For \( n = 5 \): \( 5 + 2 = 7 \)

Terms: \( 3, 4, 5, 6, 7 \)

2. \( n - 7 \)
- For \( n = 1 \): \( 1 - 7 = -6 \)
- For \( n = 2 \): \( 2 - 7 = -5 \)
- For \( n = 3 \): \( 3 - 7 = -4 \)
- For \( n = 4 \): \( 4 - 7 = -3 \)
- For \( n = 5 \): \( 5 - 7 = -2 \)

Terms: \( -6, -5, -4, -3, -2 \)

3. \( 4n \)
- For \( n = 1 \): \( 4 \cdot 1 = 4 \)
- For \( n = 2 \): \( 4 \cdot 2 = 8 \)
- For \( n = 3 \): \( 4 \cdot 3 = 12 \)
- For \( n = 4 \): \( 4 \cdot 4 = 16 \)
- For \( n = 5 \): \( 4 \cdot 5 = 20 \)

Terms: \( 4, 8, 12, 16, 20 \)

4. \( 3n + 1 \)
- For \( n = 1 \): \( 3 \cdot 1 + 1 = 4 \)
- For \( n = 2 \): \( 3 \cdot 2 + 1 = 7 \)
- For \( n = 3 \): \( 3 \cdot 3 + 1 = 10 \)
- For \( n = 4 \): \( 3 \cdot 4 + 1 = 13 \)
- For \( n = 5 \): \( 3 \cdot 5 + 1 = 16 \)

Terms: \( 4, 7, 10, 13, 16 \)

5. \( 5n - 8 \)
- For \( n = 1 \): \( 5 \cdot 1 - 8 = -3 \)
- For \( n = 2 \): \( 5 \cdot 2 - 8 = 2 \)
- For \( n = 3 \): \( 5 \cdot 3 - 8 = 7 \)
- For \( n = 4 \): \( 5 \cdot 4 - 8 = 12 \)
- For \( n = 5 \): \( 5 \cdot 5 - 8 = 17 \)

Terms: \( -3, 2, 7, 12, 17 \)

6. \( 2n + 0.5 \)
- For \( n = 1 \): \( 2 \cdot 1 + 0.5 = 2.5 \)
- For \( n = 2 \): \( 2 \cdot 2 + 0.5 = 4.5 \)
- For \( n = 3 \): \( 2 \cdot 3 + 0.5 = 6.5 \)
- For \( n = 4 \): \( 2 \cdot 4 + 0.5 = 8.5 \)
- For \( n = 5 \): \( 2 \cdot 5 + 0.5 = 10.5 \)

Terms: \( 2.5, 4.5, 6.5, 8.5, 10.5 \)

7. \( 12n - 1.5 \)
- For \( n = 1 \): \( 12 \cdot 1 - 1.5 = 10.5 \)
- For \( n = 2 \): \( 12 \cdot 2 - 1.5 = 22.5 \)
- For \( n = 3 \): \( 12 \cdot 3 - 1.5 = 34.5 \)
- For \( n = 4 \): \( 12 \cdot 4 - 1.5 = 46.5 \)
- For \( n = 5 \): \( 12 \cdot 5 - 1.5 = 58.5 \)

Terms: \( 10.5, 22.5, 34.5, 46.5, 58.5 \)

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#### Section B: Find the first five terms of the following sequences.

1. \( 2 - n \)
- For \( n = 1 \): \( 2 - 1 = 1 \)
- For \( n = 2 \): \( 2 - 2 = 0 \)
- For \( n = 3 \): \( 2 - 3 = -1 \)
- For \( n = 4 \): \( 2 - 4 = -2 \)
- For \( n = 5 \): \( 2 - 5 = -3 \)

Terms: \( 1, 0, -1, -2, -3 \)

2. \( 13 - n \)
- For \( n = 1 \): \( 13 - 1 = 12 \)
- For \( n = 2 \): \( 13 - 2 = 11 \)
- For \( n = 3 \): \( 13 - 3 = 10 \)
- For \( n = 4 \): \( 13 - 4 = 9 \)
- For \( n = 5 \): \( 13 - 5 = 8 \)

Terms: \( 12, 11, 10, 9, 8 \)

3. \( -3n \)
- For \( n = 1 \): \( -3 \cdot 1 = -3 \)
- For \( n = 2 \): \( -3 \cdot 2 = -6 \)
- For \( n = 3 \): \( -3 \cdot 3 = -9 \)
- For \( n = 4 \): \( -3 \cdot 4 = -12 \)
- For \( n = 5 \): \( -3 \cdot 5 = -15 \)

Terms: \( -3, -6, -9, -12, -15 \)

4. \( 1 - 2n \)
- For \( n = 1 \): \( 1 - 2 \cdot 1 = -1 \)
- For \( n = 2 \): \( 1 - 2 \cdot 2 = -3 \)
- For \( n = 3 \): \( 1 - 2 \cdot 3 = -5 \)
- For \( n = 4 \): \( 1 - 2 \cdot 4 = -7 \)
- For \( n = 5 \): \( 1 - 2 \cdot 5 = -9 \)

Terms: \( -1, -3, -5, -7, -9 \)

5. \( -5n + 3 \)
- For \( n = 1 \): \( -5 \cdot 1 + 3 = -2 \)
- For \( n = 2 \): \( -5 \cdot 2 + 3 = -7 \)
- For \( n = 3 \): \( -5 \cdot 3 + 3 = -12 \)
- For \( n = 4 \): \( -5 \cdot 4 + 3 = -17 \)
- For \( n = 5 \): \( -5 \cdot 5 + 3 = -22 \)

Terms: \( -2, -7, -12, -17, -22 \)

6. \( \frac{1}{2}n \)
- For \( n = 1 \): \( \frac{1}{2} \cdot 1 = 0.5 \)
- For \( n = 2 \): \( \frac{1}{2} \cdot 2 = 1 \)
- For \( n = 3 \): \( \frac{1}{2} \cdot 3 = 1.5 \)
- For \( n = 4 \): \( \frac{1}{2} \cdot 4 = 2 \)
- For \( n = 5 \): \( \frac{1}{2} \cdot 5 = 2.5 \)

Terms: \( 0.5, 1, 1.5, 2, 2.5 \)

7. \( \frac{3}{4}n \)
- For \( n = 1 \): \( \frac{3}{4} \cdot 1 = 0.75 \)
- For \( n = 2 \): \( \frac{3}{4} \cdot 2 = 1.5 \)
- For \( n = 3 \): \( \frac{3}{4} \cdot 3 = 2.25 \)
- For \( n = 4 \): \( \frac{3}{4} \cdot 4 = 3 \)
- For \( n = 5 \): \( \frac{3}{4} \cdot 5 = 3.75 \)

Terms: \( 0.75, 1.5, 2.25, 3, 3.75 \)

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#### Section C: Find the first four terms of the following sequences.

1. \( 2n - 12 \)
- For \( n = 1 \): \( 2 \cdot 1 - 12 = -10 \)
- For \( n = 2 \): \( 2 \cdot 2 - 12 = -8 \)
- For \( n = 3 \): \( 2 \cdot 3 - 12 = -6 \)
- For \( n = 4 \): \( 2 \cdot 4 - 12 = -4 \)

Terms: \( -10, -8, -6, -4 \)

2. \( 6n - 10 \)
- For \( n = 1 \): \( 6 \cdot 1 - 10 = -4 \)
- For \( n = 2 \): \( 6 \cdot 2 - 10 = 2 \)
- For \( n = 3 \): \( 6 \cdot 3 - 10 = 8 \)
- For \( n = 4 \): \( 6 \cdot 4 - 10 = 14 \)

Terms: \( -4, 2, 8, 14 \)

3. \( n - 98 \)
- For \( n = 1 \): \( 1 - 98 = -97 \)
- For \( n = 2 \): \( 2 - 98 = -96 \)
- For \( n = 3 \): \( 3 - 98 = -95 \)
- For \( n = 4 \): \( 4 - 98 = -94 \)

Terms: \( -97, -96, -95, -94 \)

4. \( 3n + 7 \)
- For \( n = 1 \): \( 3 \cdot 1 + 7 = 10 \)
- For \( n = 2 \): \( 3 \cdot 2 + 7 = 13 \)
- For \( n = 3 \): \( 3 \cdot 3 + 7 = 16 \)
- For \( n = 4 \): \( 3 \cdot 4 + 7 = 19 \)

Terms: \( 10, 13, 16, 19 \)

5. \( 2.5n \)
- For \( n = 1 \): \( 2.5 \cdot 1 = 2.5 \)
- For \( n = 2 \): \( 2.5 \cdot 2 = 5 \)
- For \( n = 3 \): \( 2.5 \cdot 3 = 7.5 \)
- For \( n = 4 \): \( 2.5 \cdot 4 = 10 \)

Terms: \( 2.5, 5, 7.5, 10 \)

6. \( 4 - 2n \)
- For \( n = 1 \): \( 4 - 2 \cdot 1 = 2 \)
- For \( n = 2 \): \( 4 - 2 \cdot 2 = 0 \)
- For \( n = 3 \): \( 4 - 2 \cdot 3 = -2 \)
- For \( n = 4 \): \( 4 - 2 \cdot 4 = -4 \)

Terms: \( 2, 0, -2, -4 \)

7. \( 8 - n \)
- For \( n = 1 \): \( 8 - 1 = 7 \)
- For \( n = 2 \): \( 8 - 2 = 6 \)
- For \( n = 3 \): \( 8 - 3 = 5 \)
- For \( n = 4 \): \( 8 - 4 = 4 \)

Terms: \( 7, 6, 5, 4 \)

8. \( 0.5n + 3 \)
- For \( n = 1 \): \( 0.5 \cdot 1 + 3 = 3.5 \)
- For \( n = 2 \): \( 0.5 \cdot 2 + 3 = 4 \)
- For \( n = 3 \): \( 0.5 \cdot 3 + 3 = 4.5 \)
- For \( n = 4 \): \( 0.5 \cdot 4 + 3 = 5 \)

Terms: \( 3.5, 4, 4.5, 5 \)

9. \( 6 - 3n \)
- For \( n = 1 \): \( 6 - 3 \cdot 1 = 3 \)
- For \( n = 2 \): \( 6 - 3 \cdot 2 = 0 \)
- For \( n = 3 \): \( 6 - 3 \cdot 3 = -3 \)
- For \( n = 4 \): \( 6 - 3 \cdot 4 = -6 \)

Terms: \( 3, 0, -3, -6 \)

10. \( 6n \)
- For \( n = 1 \): \( 6 \cdot 1 = 6 \)
- For \( n = 2 \): \( 6 \cdot 2 = 12 \)
- For \( n = 3 \): \( 6 \cdot 3 = 18 \)
- For \( n = 4 \): \( 6 \cdot 4 = 24 \)

Terms: \( 6, 12, 18, 24 \)

11. \( \frac{n}{2} \)
- For \( n = 1 \): \( \frac{1}{2} = 0.5 \)
- For \( n = 2 \): \( \frac{2}{2} = 1 \)
- For \( n = 3 \): \( \frac{3}{2} = 1.5 \)
- For \( n = 4 \): \( \frac{4}{2} = 2 \)

Terms: \( 0.5, 1, 1.5, 2 \)

12. \( 8 - 5n \)
- For \( n = 1 \): \( 8 - 5 \cdot 1 = 3 \)
- For \( n = 2 \): \( 8 - 5 \cdot 2 = -2 \)
- For \( n = 3 \): \( 8 - 5 \cdot 3 = -7 \)
- For \( n = 4 \): \( 8 - 5 \cdot 4 = -12 \)

Terms: \( 3, -2, -7, -12 \)

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#### Section D (Level 7!): Find the first three terms of the following sequences.

1. \( 2(n + 1) \)
- For \( n = 1 \): \( 2(1 + 1) = 2 \cdot 2 = 4 \)
- For \( n = 2 \): \( 2(2 + 1) = 2 \cdot 3 = 6 \)
- For \( n = 3 \): \( 2(3 + 1) = 2 \cdot 4 = 8 \)

Terms: \( 4, 6, 8 \)

2. \( n^2 \)
- For \( n = 1 \): \( 1^2 = 1 \)
- For \( n = 2 \): \( 2^2 = 4 \)
- For \( n = 3 \): \( 3^2 = 9 \)

Terms: \( 1, 4, 9 \)

3. \( \frac{n + 1}{n} \)
- For \( n = 1 \): \( \frac{1 + 1}{1} = \frac{2}{1} = 2 \)
- For \( n = 2 \): \( \frac{2 + 1}{2} = \frac{3}{2} = 1.5 \)
- For \( n = 3 \): \( \frac{3 + 1}{3} = \frac{4}{3} \approx 1.33 \)

Terms: \( 2, 1.5, \frac{4}{3} \)

4. \( 3(n + 5) \)
- For \( n = 1 \): \( 3(1 + 5) = 3 \cdot 6 = 18 \)
- For \( n = 2 \): \( 3(2 + 5) = 3 \cdot 7 = 21 \)
- For \( n = 3 \): \( 3(3 + 5) = 3 \cdot 8 = 24 \)

Terms: \( 18, 21, 24 \)

5. \( n^2 + 1 \)
- For \( n = 1 \): \( 1^2 + 1 = 1 + 1 = 2 \)
- For \( n = 2 \): \( 2^2 + 1 = 4 + 1 = 5 \)
- For \( n = 3 \): \( 3^2 + 1 = 9 + 1 = 10 \)

Terms: \( 2, 5, 10 \)

6. \( \frac{3n}{2} \)
- For \( n = 1 \): \( \frac{3 \cdot 1}{2} = \frac{3}{2} = 1.5 \)
- For \( n = 2 \): \( \frac{3 \cdot 2}{2} = \frac{6}{2} = 3 \)
- For \( n = 3 \): \( \frac{3 \cdot 3}{2} = \frac{9}{2} = 4.5 \)

Terms: \( 1.5, 3, 4.5 \)

7. \( 10(n - 9) \)
- For \( n = 1 \): \( 10(1 - 9) = 10 \cdot (-8) = -80 \)
- For \( n = 2 \): \( 10(2 - 9) = 10 \cdot (-7) = -70 \)
- For \( n = 3 \): \( 10(3 - 9) = 10 \cdot (-6) = -60 \)

Terms: \( -80, -70, -60 \)

8. \( n^2 + 5 \)
- For \( n = 1 \): \( 1^2 + 5 = 1 + 5 = 6 \)
- For \( n = 2 \): \( 2^2 + 5 = 4 + 5 = 9 \)
- For \( n = 3 \): \( 3^2 + 5 = 9 + 5 = 14 \)

Terms: \( 6, 9, 14 \)

9. \( n(n + 2) \)
- For \( n = 1 \): \( 1(1 + 2) = 1 \cdot 3 = 3 \)
- For \( n = 2 \): \( 2(2 + 2) = 2 \cdot 4 = 8 \)
- For \( n = 3 \): \( 3(3 + 2) = 3 \cdot 5 = 15 \)

Terms: \( 3, 8, 15 \)

10. \( 25(1 - n) \)
- For \( n = 1 \): \( 25(1 - 1) = 25 \cdot 0 = 0 \)
- For \( n = 2 \): \( 25(1 - 2) = 25 \cdot (-1) = -25 \)
- For \( n = 3 \): \( 25(1 - 3) = 25 \cdot (-2) = -50 \)

Terms: \( 0, -25, -50 \)

11. \( 2n^2 \)
- For \( n = 1 \): \( 2 \cdot 1^2 = 2 \cdot 1 = 2 \)
- For \( n = 2 \): \( 2 \cdot 2^2 = 2 \cdot 4 = 8 \)
- For \( n = 3 \): \( 2 \cdot 3^2 = 2 \cdot 9 = 18 \)

Terms: \( 2, 8, 18 \)

12. \( n^3 \)
- For \( n = 1 \): \( 1^3 = 1 \)
- For \( n = 2 \): \( 2^3 = 8 \)
- For \( n = 3 \): \( 3^3 = 27 \)

Terms: \( 1, 8, 27 \)

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Final Answer:

\[
\boxed{
\text{See detailed solutions above.}
}
\]