Question

The number of circles at Stage 10 is extremely large.
Stage

Number of Circles (Kiran)

2

4

8

Number of Circles (Jada)

1

2

2

Number of Circles (Noah)

2 ^ 1

2 ^ 2

2-2

3

4

5

10

2 ^ 3

16

2 ^ 4

32

2 ^ 5

1024

2 ^ 10

Write an expression to represent this number.

Answer 1

To find the total number of circles at Stage 10 for Kiran, Jada, and Noah, we can analyze the patterns in their numbers.

### Kiran's Number of Circles:
From the data, we have:
- Stage 2: 4 circles
- Stage 3: 8 circles

It looks like Kiran's number of circles is doubling each stage starting from Stage 3. Therefore, we can express Kiran's circles as:
- Kiran's circles at Stage \( n \) (for \( n \geq 3 \)):
\[ K(n) = 2^{n + 1} \]

### Jada's Number of Circles:
From the data, we have:
- Stage 2: 1 circle
- Stage 3: 2 circles
- Stage 4: 2 circles
From the data, it suggests that after stage 3, Jada has a fixed number of circles (2). Hence we can represent it as:
- Jada's circles at Stage \( n \):
\[ J(n) = \begin{cases}
1 & \text{if } n = 2 \\
2 & \text{if } n \geq 3
\end{cases} \]

### Noah's Number of Circles:
From the data, we have:
- Stage 2: \( 2^1 \) (which is 2)
- Stage 3: \( 2^2 \) (which is 4)
- Stage 4: \( 2^3 \) (which is 8)
- Stage 5: \( 2^4 \) (which is 16)
- Stage 10: \( 2^{10} \) (which is 1024)

So, Noah's circles at stage \( n \) can be expressed as:
- Noah's circles at Stage \( n \):
\[ N(n) = 2^{n - 1} \]

### Total Number of Circles at Stage 10:
Now, to express the total number of circles at Stage 10, we can use:
- \( K(10) = 2^{10 + 1} = 2048 \)
- \( J(10) = 2 \)
- \( N(10) = 2^{10 - 1} = 2^9 = 512 \)

Finally, the total number of circles at Stage 10 is given by:
\[
\text{Total Circles}(10) = K(10) + J(10) + N(10)
\]
\[
\text{Total Circles}(10) = 2048 + 2 + 512 = 2562
\]

Thus, the total number of circles at Stage 10 is **2562**.