This activity sheet helps students discover the fascinating Fibonacci sequence by continuing the pattern and exploring its connection to Pascal's triangle.

Problem Description:

The task involves two main parts:
1. Continuing the Fibonacci Series: The Fibonacci series is a sequence where each number is the sum of the two preceding ones. The given sequence starts as `0, 1, 1, 2, 3, 5, 8`. The task is to continue this series by adding the last two numbers to generate the next one.
2. Constructing Pascal's Triangle: Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The task is to construct Pascal's triangle using the given grid.

---

Solution:

#### Part 1: Continuing the Fibonacci Series
The Fibonacci series is defined as:
- \( F(n) = F(n-1) + F(n-2) \)
- Starting with \( F(0) = 0 \) and \( F(1) = 1 \).

Given sequence: \( 0, 1, 1, 2, 3, 5, 8 \)

To find the next number:
- The last two numbers are \( 5 \) and \( 8 \).
- Add them: \( 5 + 8 = 13 \).

Next number: \( 13 \)

To find the subsequent number:
- The last two numbers are now \( 8 \) and \( 13 \).
- Add them: \( 8 + 13 = 21 \).

Next number: \( 21 \)

Continuing this process:
- Last two numbers: \( 13 \) and \( 21 \)
- Add them: \( 13 + 21 = 34 \)

Next number: \( 34 \)

Thus, the extended Fibonacci series is:
\[ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 \]

---

#### Part 2: Constructing Pascal's Triangle
Pascal's triangle is constructed by starting with a single `1` at the top. Each subsequent row is generated by adding adjacent numbers from the previous row, with `1`s on the edges.

Given grid:
```
1
1 1
1 _ 1
1 _ _ 1
```

Step-by-Step Construction:

1. Row 1 (Top Row):
```
1
```

2. Row 2:
```
1 1
```

3. Row 3:
- The first and last numbers are `1`.
- The middle number is the sum of the two numbers directly above it: \( 1 + 1 = 2 \).
```
1 2 1
```

4. Row 4:
- The first and last numbers are `1`.
- The second number is the sum of the first and second numbers in Row 3: \( 1 + 2 = 3 \).
- The third number is the sum of the second and third numbers in Row 3: \( 2 + 1 = 3 \).
```
1 3 3 1
```

Thus, the completed Pascal's triangle is:
```
1
1 1
1 2 1
1 3 3 1
```

---

Final Answers:

1. Extended Fibonacci Series:
\[
\boxed{0, 1, 1, 2, 3, 5, 8, 13, 21, 34}
\]

2. Completed Pascal's Triangle:
\[
\boxed{
\begin{array}{cccc}
& & 1 & \\
& 1 & 2 & 1 \\
1 & 3 & 3 & 1 \\
\end{array}
}
\]