Answer:
(a) 51
(b) [tex]T_{n}=a+(n-1)d[/tex]
(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:
[tex]T_{n}=a+(n-1)d[/tex]
(a)
Compute the number of toothpicks required for the 10th figure as follows:
[tex]T_{n}=a+(n-1)d[/tex]
[tex]T_{10}=6+(10-1)\times5[/tex]
[tex]=6+(9\times5)\\=6+45\\=51[/tex]
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:
[tex]T_{n}=a+(n-1)d[/tex]
(c)
Compute the value of n for [tex]T_{n}=102[/tex] as follows:
[tex]T_{n}=102[/tex]
[tex]a+(n-1)d=102\\\\[/tex]
[tex]6+(n-1)\times 5=102[/tex]
[tex]5(n-1)=96[/tex]
[tex]n-1=19.2[/tex]
[tex]n=20.2\\n\approx 20[/tex]
Thus, the 20th figure will require exactly 102 toothpicks.
