Explicit Expressions And Recursive Processes Matching Worksheet ... - Free Printable

Problem Analysis:

The task involves matching explicit and recursive formulas for sequences. Here's a breakdown of the problem:

1. Explicit Formula to Recursive Formula: Convert an explicit formula into its corresponding recursive formula.
2. Recursive Formula to Explicit Formula: Convert a recursive formula into its corresponding explicit formula.

We are given six problems, each with either an explicit or recursive formula, and we need to match them correctly.

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Solution:

#### Problem 1:
- Given Recursive Formula:
\[
L_0 = 0, \quad L_n = L_{n-1} + 6
\]
- Task: Write the explicit formula for the sequence.
- Solution:
The recursive formula indicates that the sequence starts at 0 and increases by 6 for each subsequent term. This is an arithmetic sequence with the first term \( L_0 = 0 \) and common difference \( d = 6 \). The explicit formula for an arithmetic sequence is:
\[
L_n = L_0 + n \cdot d
\]
Substituting \( L_0 = 0 \) and \( d = 6 \):
\[
L_n = 0 + n \cdot 6 = 6n
\]
Answer: \( L_n = 6n \)

#### Problem 2:
- Given Explicit Formula:
\[
L_n = 4n - 1
\]
- Task: Write the recursive formula for the sequence.
- Solution:
The explicit formula \( L_n = 4n - 1 \) shows that the sequence is linear, with a slope of 4. This means each term increases by 4 from the previous term. The first term is:
\[
L_0 = 4(0) - 1 = -1
\]
The recursive formula is:
\[
L_0 = -1, \quad L_n = L_{n-1} + 4
\]
Answer: \( L_0 = -1, \quad L_n = L_{n-1} + 4 \)

#### Problem 3:
- Given Recursive Formula:
\[
L_0 = 5, \quad L_n = L_{n-1} + 7
\]
- Task: Write the explicit formula for the sequence.
- Solution:
The recursive formula indicates that the sequence starts at 5 and increases by 7 for each subsequent term. This is an arithmetic sequence with the first term \( L_0 = 5 \) and common difference \( d = 7 \). The explicit formula for an arithmetic sequence is:
\[
L_n = L_0 + n \cdot d
\]
Substituting \( L_0 = 5 \) and \( d = 7 \):
\[
L_n = 5 + n \cdot 7 = 5 + 7n
\]
Answer: \( L_n = 5 + 7n \)

#### Problem 4:
- Given Sequence:
\[
3, 8, 13, 18, \ldots
\]
- Task: Write a recursive formula for the sequence.
- Solution:
The sequence is arithmetic, with a common difference of 5. The first term is \( L_0 = 3 \). The recursive formula is:
\[
L_0 = 3, \quad L_n = L_{n-1} + 5
\]
Answer: \( L_0 = 3, \quad L_n = L_{n-1} + 5 \)

#### Problem 5:
- Given Recursive Formula:
\[
L_0 = 0, \quad L_n = L_{n-1} + 9
\]
- Task: Write the explicit formula for the sequence.
- Solution:
The recursive formula indicates that the sequence starts at 0 and increases by 9 for each subsequent term. This is an arithmetic sequence with the first term \( L_0 = 0 \) and common difference \( d = 9 \). The explicit formula for an arithmetic sequence is:
\[
L_n = L_0 + n \cdot d
\]
Substituting \( L_0 = 0 \) and \( d = 9 \):
\[
L_n = 0 + n \cdot 9 = 9n
\]
Answer: \( L_n = 9n \)

#### Problem 6:
- Given Explicit Formula:
\[
L_n = 7n - 3
\]
- Task: Write the recursive formula for the sequence.
- Solution:
The explicit formula \( L_n = 7n - 3 \) shows that the sequence is linear, with a slope of 7. This means each term increases by 7 from the previous term. The first term is:
\[
L_0 = 7(0) - 3 = -3
\]
The recursive formula is:
\[
L_0 = -3, \quad L_n = L_{n-1} + 7
\]
Answer: \( L_0 = -3, \quad L_n = L_{n-1} + 7 \)

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Final Matching:

1. \( L_n = 6n \) → a. \( 6(n-1) \)
2. \( L_0 = -1, \quad L_n = L_{n-1} + 4 \) → b. \( 4(n-1) \)
3. \( L_n = 5 + 7n \) → c. \( L_0 = 5, \quad L_n = L_{n-1} + 7 \)
4. \( L_0 = 3, \quad L_n = L_{n-1} + 5 \) → d. \( 5(n-1) \)
5. \( L_n = 9n \) → e. \( 9(n-1) \)
6. \( L_0 = -3, \quad L_n = L_{n-1} + 7 \) → f. \( L_0 = 3, \quad L_n = L_{n-1} + 9 \)

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Final Answer:

\[
\boxed{a, b, c, d, e, f}
\]