Matching worksheet for writing explicit and recursive formulas for sequences.

Let’s solve this “Matching Worksheet” step by step. The goal is to match each problem (1–6) with the correct lettered answer (a–f) by converting between explicit and recursive formulas for sequences.

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🔢 Problem 1:

> Given the recursive formula, write the explicit formula for the sequence:
> t₁ = 0
> tₙ = tₙ₋₁ + 4

This is an arithmetic sequence starting at 0, increasing by 4 each time.

- Recursive: tₙ = tₙ₋₁ + 4
- Explicit: tₙ = 0 + 4(n - 1) = 4(n - 1)

✔ Match: b. 4(n - 1)

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🔢 Problem 2:

> Given the explicit formula, write the recursive formula for the sequence:
> tₙ = 4n - 1

First, find the first term:
t₁ = 4(1) - 1 = 3

Find second term: t₂ = 4(2) - 1 = 7 → difference = 4
So it’s arithmetic with common difference 4.

Recursive formula:
t₁ = 3
tₙ = tₙ₋₁ + 4

✔ Match: e. t₁ = 3, tₙ = tₙ₋₁ + 4

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🔢 Problem 3:

> Given the recursive formula, write the explicit formula for the sequence:
> t₁ = 0
> tₙ = tₙ₋₁ - 6

Arithmetic sequence starting at 0, decreasing by 6.

Explicit: tₙ = 0 + (-6)(n - 1) = -6(n - 1)

✔ Match: d. -6(n - 1)

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🔢 Problem 4:

> Given the recursive formula for the following sequence:
> 3, 8, 13, 18, ...

First, observe the pattern:
8 - 3 = 5, 13 - 8 = 5, 18 - 13 = 5 → common difference = 5

So recursive formula:
t₁ = 3
tₙ = tₙ₋₁ + 5

✔ Match: f. tₙ = tₙ₋₁ + 5

*(Note: This is asking for recursive formula given sequence — so we write the recursive form.)*

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🔢 Problem 5:

> Given the recursive formula, write the explicit formula for the sequence:
> t₁ = 0
> tₙ = tₙ₋₁ - 9

Arithmetic sequence, starting at 0, decreasing by 9.

Explicit: tₙ = 0 + (-9)(n - 1) = -9(n - 1)

✔ Match: a. -9(n - 1)

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🔢 Problem 6:

> Given the explicit formula, write the recursive formula for the sequence:
> tₙ = 7n - 1

First term: t₁ = 7(1) - 1 = 6
Second term: t₂ = 7(2) - 1 = 13 → difference = 7

So recursive:
t₁ = 6
tₙ = tₙ₋₁ + 7

✔ Match: c. t₁ = 6, tₙ = tₙ₋₁ + 7

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## ✔ Final Matching Answers:

1. → b
2. → e
3. → d
4. → f
5. → a
6. → c

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## 📝 Summary Table:

| Problem | Answer |
|---------|--------|
| 1 | b |
| 2 | e |
| 3 | d |
| 4 | f |
| 5 | a |
| 6 | c |

You can now fill in the worksheet with these letters!

Let me know if you’d like a visual explanation or more examples!