Question

The triangular figures shown below are constructed with toothpicks. The pattern shows what happens with 1 triangle, with 2 triangles, and with 3 triangles. Describe a possible pattern. Assuming the pattern continues, answer the following questions. How many toothpicks would be needed for the 10th figure? How many toothpicks are needed for the nth figure? In which figure are exactly 102 toothpicks used?

Answer 1

Answer:

(a) 51

(b)  $T_{n}=a+(n-1)d$

(c) 20th figure

Step-by-step explanation:

The diagram is attached below.

The first triangle is formed using 6 toothpicks.

The second triangle is formed using 11 toothpicks.

The third triangle is formed using 16 toothpicks.

So, there is an increases of 5 toothpicks every time.

We can say that the number of toothpicks used to form n triangles are following an arithmetic progression with the first term as 6 and the common difference as 5.

The nth term will be:

$T_{n}=a+(n-1)d$

(a)

Compute the number of toothpicks required for the 10th figure as follows:

$T_{n}=a+(n-1)d$

$T_{10}=6+(10-1)\times5$

     $=6+(9\times5)\\=6+45\\=51$

Thus, the number of toothpicks required for the 10th figure is 51.

(b)

Compute the number of toothpicks required for the nth figure as follows:

$T_{n}=a+(n-1)d$

(c)

Compute the value of n for $T_{n}=102$ as follows:

                 $T_{n}=102$

   $a+(n-1)d=102$

$6+(n-1)\times 5=102$

         $5(n-1)=96$

             $n-1=19.2$

                   $n=20.2\\n\approx 20$

Thus, the 20th figure will require exactly 102 toothpicks.