Algebra 2 Worksheets | Sequences and Series Worksheets - Free Printable

Problem Analysis:

The task involves analyzing sequences and determining the next three terms, as well as providing explicit and recursive formulas for each sequence. Additionally, we need to generate the first four terms of given recursive formulas.

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Part 1: Analyze Sequences and Provide Formulas

#### Sequence 1: \( 46, 45, 44, 43 \)

- Pattern: Each term decreases by 1.
- Next 3 Terms:
\[
42, 41, 40
\]
- Recursive Formula:
\[
a_n = a_{n-1} - 1, \quad a_1 = 46
\]
- Explicit Formula:
\[
a_n = 46 - (n - 1) = 47 - n
\]

#### Sequence 2: \( 3, 9, 27, 81 \)

- Pattern: Each term is multiplied by 3.
- Next 3 Terms:
\[
243, 729, 2187
\]
- Recursive Formula:
\[
a_n = a_{n-1} \cdot 3, \quad a_1 = 3
\]
- Explicit Formula:
\[
a_n = 3^n
\]

#### Sequence 3: \( 47, 35, 15, -13 \)

- Pattern: The differences between consecutive terms are:
\[
35 - 47 = -12, \quad 15 - 35 = -20, \quad -13 - 15 = -28
\]
The differences increase by \(-8\) each time.
- Next 3 Terms:
\[
-13 - 36 = -49, \quad -49 - 44 = -93, \quad -93 - 52 = -145
\]
- Recursive Formula:
\[
a_n = a_{n-1} - (8(n-1) + 4), \quad a_1 = 47
\]
- Explicit Formula:
\[
a_n = 47 - 4(n-1) - 4(n-1)(n-2)
\]
Simplifying:
\[
a_n = 47 - 4(n-1) - 4(n^2 - 3n + 2) = 47 - 4n + 4 - 4n^2 + 12n - 8 = -4n^2 + 8n + 33
\]

#### Sequence 4: \( 4, 8, 12, 16 \)

- Pattern: Each term increases by 4.
- Next 3 Terms:
\[
20, 24, 28
\]
- Recursive Formula:
\[
a_n = a_{n-1} + 4, \quad a_1 = 4
\]
- Explicit Formula:
\[
a_n = 4 + 4(n-1) = 4n
\]

#### Sequence 5: \( 35, 33, 31, 29 \)

- Pattern: Each term decreases by 2.
- Next 3 Terms:
\[
27, 25, 23
\]
- Recursive Formula:
\[
a_n = a_{n-1} - 2, \quad a_1 = 35
\]
- Explicit Formula:
\[
a_n = 35 - 2(n-1) = 37 - 2n
\]

#### Sequence 6: \( 8, 16, 32, 64 \)

- Pattern: Each term is multiplied by 2.
- Next 3 Terms:
\[
128, 256, 512
\]
- Recursive Formula:
\[
a_n = a_{n-1} \cdot 2, \quad a_1 = 8
\]
- Explicit Formula:
\[
a_n = 8 \cdot 2^{n-1} = 2^3 \cdot 2^{n-1} = 2^{n+2}
\]

---

Part 2: Generate First 4 Terms from Recursive Formulas

#### Sequence 7: \( a_n = a_{n-1} \cdot 2, \, a_1 = 2 \)

- Terms:
\[
a_1 = 2, \quad a_2 = 2 \cdot 2 = 4, \quad a_3 = 4 \cdot 2 = 8, \quad a_4 = 8 \cdot 2 = 16
\]
- First 4 Terms:
\[
2, 4, 8, 16
\]

#### Sequence 8: \( a_n = a_{n-1} - 1, \, a_1 = 95 \)

- Terms:
\[
a_1 = 95, \quad a_2 = 95 - 1 = 94, \quad a_3 = 94 - 1 = 93, \quad a_4 = 93 - 1 = 92
\]
- First 4 Terms:
\[
95, 94, 93, 92
\]

#### Sequence 9: \( a_n = a_{n-1} - 4, \, a_1 = 0 \)

- Terms:
\[
a_1 = 0, \quad a_2 = 0 - 4 = -4, \quad a_3 = -4 - 4 = -8, \quad a_4 = -8 - 4 = -12
\]
- First 4 Terms:
\[
0, -4, -8, -12
\]

#### Sequence 10: \( a_n = a_{n-1} - 6, \, a_1 = -7 \)

- Terms:
\[
a_1 = -7, \quad a_2 = -7 - 6 = -13, \quad a_3 = -13 - 6 = -19, \quad a_4 = -19 - 6 = -25
\]
- First 4 Terms:
\[
-7, -13, -19, -25
\]

#### Sequence 11: \( a_n = a_{n-1} - 4, \, a_1 = 49 \)

- Terms:
\[
a_1 = 49, \quad a_2 = 49 - 4 = 45, \quad a_3 = 45 - 4 = 41, \quad a_4 = 41 - 4 = 37
\]
- First 4 Terms:
\[
49, 45, 41, 37
\]

#### Sequence 12: \( a_n = a_{n-1} - 2, \, a_1 = 64 \)

- Terms:
\[
a_1 = 64, \quad a_2 = 64 - 2 = 62, \quad a_3 = 62 - 2 = 60, \quad a_4 = 60 - 2 = 58
\]
- First 4 Terms:
\[
64, 62, 60, 58
\]

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Final Answers

#### Part 1: Sequences
1. Next 3 terms: \( 42, 41, 40 \)
- Recursive: \( a_n = a_{n-1} - 1, \, a_1 = 46 \)
- Explicit: \( a_n = 47 - n \)

2. Next 3 terms: \( 243, 729, 2187 \)
- Recursive: \( a_n = a_{n-1} \cdot 3, \, a_1 = 3 \)
- Explicit: \( a_n = 3^n \)

3. Next 3 terms: \( -49, -93, -145 \)
- Recursive: \( a_n = a_{n-1} - (8(n-1) + 4), \, a_1 = 47 \)
- Explicit: \( a_n = -4n^2 + 8n + 33 \)

4. Next 3 terms: \( 20, 24, 28 \)
- Recursive: \( a_n = a_{n-1} + 4, \, a_1 = 4 \)
- Explicit: \( a_n = 4n \)

5. Next 3 terms: \( 27, 25, 23 \)
- Recursive: \( a_n = a_{n-1} - 2, \, a_1 = 35 \)
- Explicit: \( a_n = 37 - 2n \)

6. Next 3 terms: \( 128, 256, 512 \)
- Recursive: \( a_n = a_{n-1} \cdot 2, \, a_1 = 8 \)
- Explicit: \( a_n = 2^{n+2} \)

#### Part 2: First 4 Terms
7. \( 2, 4, 8, 16 \)
8. \( 95, 94, 93, 92 \)
9. \( 0, -4, -8, -12 \)
10. \( -7, -13, -19, -25 \)
11. \( 49, 45, 41, 37 \)
12. \( 64, 62, 60, 58 \)

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