Question

find a rule that gives the number of matchsticks required to make the following pattern

Answer 1

We can see that, In the first case, no.of match sticks used is 5, 
In the second case no. of match sticks used is 9,
In the third case no. of match sticks used is 13,

We can write 5 as 2*2+1,
We can write 9 as 2*4+1,
We can write 13 as 2*6+1, 

We can see that here it follows a pattern that is, 2*even number +1,

So the answer to your question is 2*n+1 Where n = [2,∞],
and remember that n ∈ Even numbers,

Hope you understand,  Have a great day !