Comprehensive math worksheet designed to help students practice calculating terms within the Fibonacci sequence using various formulas.

Let's solve each problem step by step.

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Problem 1: Find out the next terms in the following Fibonacci style sequence.

#### a) Sequence: 2, 1, 3, 4, 7, __, __, __
The rule is \( F_n = F_{n-1} + F_{n-2} \).

- \( F_5 = F_4 + F_3 = 4 + 3 = 7 \) (already given)
- \( F_6 = F_5 + F_4 = 7 + 4 = 11 \)
- \( F_7 = F_6 + F_5 = 11 + 7 = 18 \)
- \( F_8 = F_7 + F_6 = 18 + 11 = 29 \)

So, the next terms are: 11, 18, 29.

#### b) Sequence: 4, 8, 12, 20, 32, __, __, __
The rule is \( F_n = F_{n-1} + F_{n-2} \).

- \( F_5 = F_4 + F_3 = 20 + 12 = 32 \) (already given)
- \( F_6 = F_5 + F_4 = 32 + 20 = 52 \)
- \( F_7 = F_6 + F_5 = 52 + 32 = 84 \)
- \( F_8 = F_7 + F_6 = 84 + 52 = 136 \)

So, the next terms are: 52, 84, 136.

#### c) Sequence: 0, 12, 24, 36, __, __, __
The rule is \( F_n = F_{n-1} + F_{n-2} \).

- \( F_4 = F_3 + F_2 = 36 + 24 = 60 \)
- \( F_5 = F_4 + F_3 = 60 + 36 = 96 \)
- \( F_6 = F_5 + F_4 = 96 + 60 = 156 \)
- \( F_7 = F_6 + F_5 = 156 + 96 = 252 \)

So, the next terms are: 60, 96, 156.

#### d) Sequence: 13, 7, 20, 27, 47, __, __, __
The rule is \( F_n = F_{n-1} + F_{n-2} \).

- \( F_5 = F_4 + F_3 = 27 + 20 = 47 \) (already given)
- \( F_6 = F_5 + F_4 = 47 + 27 = 74 \)
- \( F_7 = F_6 + F_5 = 74 + 47 = 121 \)
- \( F_8 = F_7 + F_6 = 121 + 74 = 195 \)

So, the next terms are: 74, 121, 195.

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Problem 2: Find out the 16th Fibonacci number (\( F_{16} \)) when \( F_{13} = 233 \) and \( F_{14} = 377 \).

Using the Fibonacci rule \( F_n = F_{n-1} + F_{n-2} \):

- \( F_{15} = F_{14} + F_{13} = 377 + 233 = 610 \)
- \( F_{16} = F_{15} + F_{14} = 610 + 377 = 987 \)

So, the 16th Fibonacci number is: 987.

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Problem 3: Use the Fibonacci Sequence to complete the following diagram.

The diagram shows a rectangle divided into smaller squares, which is a visual representation of the Fibonacci sequence. The side lengths of the squares correspond to Fibonacci numbers. Starting with the smallest square (1x1), the next squares have side lengths 1, 2, 3, 5, 8, etc.

The diagram shows:
- A 1x1 square
- A 1x1 square next to it
- A 2x2 square below the two 1x1 squares
- A 3x3 square to the right of the 2x2 square

The next squares would be:
- A 5x5 square below the 3x3 square
- An 8x8 square to the right of the 5x5 square

So, the completed diagram would have squares with side lengths: 1, 1, 2, 3, 5, 8.

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Problem 4: Determine the following values in the Fibonacci Sequence.

The rule given is \( F_{-n} = (-1)^{n+1} F_n \).

#### a) \( F_{-6} \)
We know \( F_6 = 8 \) (from the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8).
Using the rule:
\[ F_{-6} = (-1)^{6+1} F_6 = (-1)^7 \cdot 8 = -8 \]

So, \( F_{-6} = -8 \).

#### b) \( F_{-8} \)
We know \( F_8 = 21 \) (from the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21).
Using the rule:
\[ F_{-8} = (-1)^{8+1} F_8 = (-1)^9 \cdot 21 = -21 \]

So, \( F_{-8} = -21 \).

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Problem 5: Find out the sum of the first 13 terms.

The formula given is:
\[ \sum_{n=0}^{N} F_n = F_{N+2} - F_2 \]

For \( N = 13 \):
\[ \sum_{n=0}^{13} F_n = F_{15} - F_2 \]

We know:
- \( F_{15} = 610 \) (from Problem 2)
- \( F_2 = 1 \) (from the Fibonacci sequence: 0, 1, 1, 2, ...)

So:
\[ \sum_{n=0}^{13} F_n = 610 - 1 = 609 \]

The sum of the first 13 terms is: 609.

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Problem 6: Complete the pattern using the Fibonacci Sequence (up to 7 terms).

#### a) Sequence: 7, 9, 16, __, __, __, __
The rule is \( F_n = F_{n-1} + F_{n-2} \).

- \( F_3 = F_2 + F_1 = 9 + 7 = 16 \) (already given)
- \( F_4 = F_3 + F_2 = 16 + 9 = 25 \)
- \( F_5 = F_4 + F_3 = 25 + 16 = 41 \)
- \( F_6 = F_5 + F_4 = 41 + 25 = 66 \)
- \( F_7 = F_6 + F_5 = 66 + 41 = 107 \)

So, the next terms are: 25, 41, 66, 107.

#### b) Sequence: 2, 14, 16, __, __, __, __
The rule is \( F_n = F_{n-1} + F_{n-2} \).

- \( F_3 = F_2 + F_1 = 14 + 2 = 16 \) (already given)
- \( F_4 = F_3 + F_2 = 16 + 14 = 30 \)
- \( F_5 = F_4 + F_3 = 30 + 16 = 46 \)
- \( F_6 = F_5 + F_4 = 46 + 30 = 76 \)
- \( F_7 = F_6 + F_5 = 76 + 46 = 122 \)

So, the next terms are: 30, 46, 76, 122.

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Final Answers:

1. a) 11, 18, 29
b) 52, 84, 136
c) 60, 96, 156
d) 74, 121, 195

2. 987

3. The completed diagram has squares with side lengths: 1, 1, 2, 3, 5, 8.

4. a) -8
b) -21

5. 609

6. a) 25, 41, 66, 107
b) 30, 46, 76, 122

\boxed{987}