Let’s write the number of match sticks in n-th position (n is a positive integer) in each arrangement:

i.

ii.

iii.

i)In first position, there are 6 sticks after that 5 sticks increase in each position.

Therefore,

In second position, there are (6 + 5) sticks

In third position, there are (6 + 5 + 5 = 6 + 5(2)) sticks

Similarly, in nth position there are (6 + (n – 1)5) = (6+5n-5)

= (1+5n) sticks

ii) In first position, there are 7 sticks after that 5 sticks increase in each position.

Therefore,

In second position, there are (7 + 5) sticks

In third position, there are (7 + 5 + 5 = 7 + 5(2)) sticks

Similarly, in nth position there are (7 + (n – 1)5) = (7 + 5n – 5)

= (2 + 5n) sticks

iii) In first position, there are 5 sticks after that 4 sticks increase in each position.

Therefore,

In second position, there are (5 + 4) sticks

In third position, there are (5 + 4 + 4 = 5 + 4(2)) sticks

Similarly, in nth position there are (5 + (n – 1)4) = (5 + 4n – 4)

= (1 + 4n) sticks