Investigate how the pattern of circles grows and answer the questions below.

For Picture $n$, the bottom row has $n$ circles.

For Picture 5, the bottom row will have 5 circles.
There will be $\boxed{5}$ circles in the bottom row of Picture 5.

For Picture $n$, the second row from the bottom has $n-1$ circles.

For Picture 5, the second row from the bottom will have $5-1=4$ circles.
There will be $\boxed{4}$ circles in the second row from the bottom of Picture 5.

For Picture $n$, the third row from the bottom has $n-2$ circles.

For Picture 5, the third row from the bottom will have $5-2=3$ circles.
There will be $\boxed{3}$ circles in the third row from the bottom of Picture 5.

For any Picture $n \ge 2$, the second row from the top always has 2 circles.

For Picture 5, the second row from the top will have 2 circles.
There will be $\boxed{2}$ circles in the second row from the top of Picture 5.

For any Picture $n \ge 1$, the top row always has 1 circle.

For Picture 5, the top row will have 1 circle.
There will be $\boxed{1}$ circle in the top row of Picture 5.

The total number of circles in Picture $n$ is the sum of the first $n$ natural numbers, given by the formula $T_n = \frac{n(n+1)}{2}$.

For Picture 5, $n=5$.

Alternatively, sum the circles in each row: $1+2+3+4+5 = 15$.
The total number of circles in Picture 5 is $\boxed{15}$.

For Picture 6, $n=6$.

You will need $\boxed{21}$ circles for Picture 6.

Activity 4

Given: Input 1 gives Output 8.
Given: Input 3 gives an unknown Output.
Given: An unknown Input gives Output 16.
Given: An unknown Input gives Output 32.
The sequence of outputs is 8, 16, 24, 32. This is an arithmetic sequence where each term is 8 times its position.
Assuming the inputs are consecutive integers starting from 1, the rule is "multiply by 8".
Let the Input be $x$. The Rule is $x \times 8$.

• 4.1 (Input for Output 16): If Output is 16, then $x \times 8 = 16$. $$x = \frac{16}{8} = 2$$ The missing input is $\boxed{2}$.
• 4.2 (Output for Input 3): If Input is 3, then Output $= 3 \times 8 = 24$. The missing output is $\boxed{24}$.
• 4.3 (Input for Output 32): If Output is 32, then $x \times 8 = 32$. $$x = \frac{32}{8} = 4$$ The missing input is $\boxed{4}$.
• 4.4 (The Rule): The rule is $\boxed{\text{multiply by 8}}$.