### Theory:

Suppose you are given with $$3$$ Matchsticks and asking you to form the letter $$C$$ using that. It is easy for you to form the letter $$C$$ right.

Your work might be like this.

Suppose we want one more $$C$$. Then how many sticks you will need?

Yes, We need another $$3$$ sticks. In total, we need $$3 + 3 = 6$$ sticks.

Adding one more letter $$C$$ will become:

Here we used $$3+3+3 = 9$$ sticks.

Thus, to form one $$C$$ we need $$3$$ sticks, to form two $$C$$'s we need $$6$$ sticks, to form three $$C$$'s we need $$9$$ sticks, and so on.

Now tabulate the details and look for the pattern.

 Number of $$C$$'s formed $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ Number of matchsticks needed $$3$$ $$6$$ $$9$$ $$12$$ …

Look at the numbers of matchsticks used to form $$C$$'s.

$$3, 6, 9, 12,….$$

In this sequence, the next number should be $$15$$.

Because the number of matchsticks needed is three times the number of $$C$$'s formed.

Let us take the general number $$n$$ for the number of matchsticks needed.

If one $$C$$ is made, then $$n=1$$.

If two $$C$$'s are made, then $$n=2$$.

If three $$C$$'s are made, then $$n=3$$.

Thus, the alphabet $$n$$ can be any natural number $$1, 2, 3, 4,...$$.

The number of matchsticks required for forming any number of $$C$$'s $$= 3 × n$$.

Instead of writing $$3 × n$$, we can write $$3n$$.

Let us try to get the answer from the above form.

To form one $$C$$, $$n=1$$ and the number of matchsticks required $$=3×1 =3$$.

To form two $$C$$, $$n=2$$ and the number of matchsticks required $$=3×2 =6$$.

To form three $$C$$, $$n=3$$ and the number of matchsticks required $$=3×3 =9$$.

Thus, we got the generalized rule to find the number of matchsticks required to form any number of $$C$$'s.
Is it possible to find the number of matchsticks required to form ten $$C$$'s without drawing its pattern?

Think!

Of course. It is possible.

Just by substituting $$n=10$$ in the rule $$3n$$, we are able to get the number of matchsticks required to form ten $$C$$'s.

That is, $$3×10 = 30$$ matchsticks required.

Now can you find the number of matchsticks required to form $$100$$ $$C$$'s?

Yes, we can get. Substitute $$n = 100$$ in the rule $$3n$$.

That is, $$3×10 = 30$$ matchsticks required.
From the above demonstration, we got the rule to find the number of matchsticks required to make a pattern of $$C$$s.

The rule is:
Number of matchsticks required = 3n
Where $$n$$ is the number of $$C$$s in the pattern and $$n$$ takes the values from the natural number $$1,2,3,4,...$$.
Important!
What is the variable?

In the example, $$n$$ is a variable. It is not a fixed value. In the above example, the variable $$n$$ can take any natural number $$1,2,,3,...$$. We can write the number of matchsticks required using the variable $$n$$.
The word 'variable' means something that can vary, i.e. change. The value of a variable is not fixed. It can take different value.
Additionally, imagine other letters of the alphabets(other than $$C$$) that can be made from matchsticks/Ice candy sticks.