A pattern of match sticks is shown below. Stage 1 has 5 match sticks. Each subsequent stage has 4 more match sticks than the previous stage. Find the rule for the number of match sticks at stage n, an

Observe the given values for "Number of match sticks" (output) corresponding to "Stage" (input):

• Stage 1 gives 5 match sticks
• Stage 2 gives 9 match sticks
• Stage 3 gives 13 match sticks
• Stage 4 gives 17 match sticks

Calculate the difference between consecutive output values:
$9 - 5 = 4$
$13 - 9 = 4$
$17 - 13 = 4$
The common difference is 4. This indicates a linear relationship, which can be expressed as $T_n = an + b$, where $a$ is the common difference.
So, the rule is of the form $T_n = 4n + b$.
Using the first pair (Stage 1, 5 match sticks):
$5 = 4(1) + b$
$5 = 4 + b$
$b = 1$
The rule for the number of match sticks at Stage $n$ is $T_n = 4n + 1$.

The table asks for the number of match sticks at Stage 5.
Using the rule $T_n = 4n + 1$ for $n=5$:

$$T_5 = 4(5) + 1 = 20 + 1 = 21$$

The missing value for 1.4.1 is 21.

The table asks for the Stage number when the number of match sticks is 41.
Set $T_n = 41$ and solve for $n$:

$$4n + 1 = 41$$ $$4n = 41 - 1$$ $$4n = 40$$ $$n = \frac{40}{4}$$ $$n = 10$$

The missing value for 1.4.2 is 10.

The completed table is:

Stage	1	2	3	4	5	10
Number of match sticks	5	9	13	17	21	41

The missing input value is $\boxed{10}$.
The missing output value is $\boxed{21}$.