Question

Observe the following pattern of overlapping triangles made with matchsticks and complete the table. Also, write the general rule that gives the number of matchsticks

Number of triangles	1	2	3	4	5	6	….	….
Number of matchsticks	6	10	14	….	….	….	….	….

Answer 1

Hint: From the table, you can see that the number of triangles is in A.P. and the number of matchsticks is also in A.P. The number of triangles has a common difference as 1 and number of matchsticks has a common difference of 4. Now, in the question, we have to predict the numbers of matchsticks that are needed when 4 triangles are overlapped which is calculated by adding 4 to the number of matchsticks when 3 triangles are overlapped. Similarly, you can find the number of matchsticks for 5 and 6 number of triangles also. The general rule which gives the number of matchsticks is the ${{n}^{th}}$ term of an A.P. of the matchsticks.

Complete step by step answer:
In the above question, we have given three types of triangles which are constructed by using matchsticks as follows:
 

And we are asked to fill the following table and also give the general rule for the relationship between the number of triangles and the matchsticks.

Number of triangles	1	2	3	4	5	6	….	….
Number of matchsticks	6	10	14	….	….	….	….	….

As you can see, the number of matchsticks corresponding to different numbers of triangles is in A.P. The first term (a) of this A.P. is 6 and common difference (d) is equal to 4. Common difference is calculated by subtracting any term from its successor term. Let us pick a term 10 and the successor of this term is 14 so the difference between them is:
$14-10=4$
This is how; we got the common difference as 4.
Now, to get the number of matchsticks corresponding to 4 triangles we are going to add 4 to 14 so the number of matchsticks corresponding to 4 triangles is equal to:
$14+4=18$

Number of triangles	1	2	3	4	5	6	….	….
Number of matchsticks	6	10	14	18	….	….	….	….

Similarly, to get the number of matchsticks corresponding to 5 triangles we are going to add 4 to 18.
$18+4=22$

Number of triangles	1	2	3	4	5	6	….	….
Number of matchsticks	6	10	14	18	22	….	….	….

Now, to get the number of matchsticks corresponding to 6 triangles we are going to add 4 to 22.
$22+4=26$

Number of triangles	1	2	3	4	5	6	….	….
Number of matchsticks	6	10	14	18	22	26	….	….

We are also asked to find the general rule that gives the number of matchsticks which is equal to the ${{n}^{th}}$ term of an A.P.
We know that ${{n}^{th}}$ term of an A.P. is equal to:
\[{{T}_{n}}=a+\left( n-1 \right)d\]
Substituting “a” as 6 and “d” as 4 in the above equation we get,
\[\begin{align}
  & {{T}_{n}}=6+\left( n-1 \right)4 \\
 & \Rightarrow {{T}_{n}}=6+4n-4 \\
 & \Rightarrow {{T}_{n}}=2+4n \\
\end{align}\]
In the above formula, “n” represents the number of triangles.

Note: We can check the number of matchsticks that we have written for 4, 5 and 6 triangles by substituting the value of “n” as 4, 5 and 6 in the general rule that we have found above.
The general rule that we have found above is equal to:
\[{{T}_{n}}=2+4n\]
To get the number of matchsticks for 4 triangles, we are going to substitute the value of n as 4 in the above equation.
\[\begin{align}
  & {{T}_{4}}=2+4\left( 4 \right) \\
 & \Rightarrow {{T}_{4}}=2+16=18 \\
\end{align}\]
As you can see, the number of matchsticks for 4 triangles which we are getting from the above rule is the same as what we have calculated. This means that we have written the correct number of matchsticks corresponding to 4 triangles.
Similarly, you can check for 5 and 6 triangles and then verify your answer.