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.
 
1.png
 
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.
 
2.png
 
Adding one more letter \(C\) will become:
 
3.png
 
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.