A pattern is shown in the table and flow diagram. Complete the flow diagram and the table.

Observe the given input and output values:

• Input 1 gives Output 5
• Input 2 gives Output 9
• Input 3 gives Output 13

Calculate the difference between consecutive output values:
$9 - 5 = 4$
$13 - 9 = 4$
The common difference is 4. This indicates a linear relationship of the form $T_n = an + b$, where $a$ is the common difference.
So, $T_n = 4n + b$.
Using the first pair (Input 1, Output 5):
$5 = 4(1) + b$
$5 = 4 + b$
$b = 1$
The rule is $T_n = 4n + 1$.

The rule for the flow diagram is $4n + 1$.
For the input value 20, calculate the output:

$$T_{20} = 4(20) + 1 = 80 + 1 = 81$$

The completed flow diagram is:
Input values:
1 $\rightarrow$
2 $\rightarrow$
3 $\rightarrow$
20 $\rightarrow$

Middle box: $\boxed{4n+1}$

Output values:
$\rightarrow$ 5
$\rightarrow$ 9
$\rightarrow$ 13
$\rightarrow$ $\boxed{81}$

Using the rule $T_n = 4n + 1$:
For Stage 1: $T_1 = 4(1) + 1 = 5$
For Stage 2: $T_2 = 4(2) + 1 = 9$
For Stage 3: $T_3 = 4(3) + 1 = 13$
For Stage 4: $T_4 = 4(4) + 1 = 17$
For Stage 5: $T_5 = 4(5) + 1 = 21$

The completed table is:

Stage	1	2	3	4	5
Number of match sticks	5	9	13	17	21

The missing values are:

• The rule in the flow diagram: $\boxed{4n+1}$
• The output for input 20: $\boxed{81}$
• The number of match sticks for Stage 5: $\boxed{21}$