Chapter 11.5, Problem 46PPS

(a)

To determine

Write a sequence of the total number of triangles in the first six figures.

Answer to Problem 46PPS

The sequence is $1,4,10,19,31,46$

Explanation of Solution

Given:

  

Write a sequence of the total number of triangles in the first six figures.

Concept Used:

The number of Triangles in First figure is 1 and in second figure is 4 and in third figure is 10.

Calculation:

  $\begin{array}{l}{a}_1=1,a_2=4,a_3=10\\ 4=1+3(2-1)\\ 10=4+3(3-1)\\ a_n=a_{n-1}+3(n-1)\end{array}$

  $\begin{array}{l}\text{Therefore:}\\ a_4=a_3+3(n-1)=10+3(4-1)=19\\ a_5=a_4+3(n-1)=19+3(5-1)=31\\ a_6=a_5+3(n-1)=31+3(6-1)=46\end{array}$

Thus, the sequence is $1,4,10,19,31,46$

(b)

To determine

Write a recursive formula for the number of triangles.

Answer to Problem 46PPS

The recursive formula is $a_n=a_{n-1}+3(n-1)$

Explanation of Solution

Given:

  

Write a recursive formula for the number of triangles.

Concept Used:

The number of Triangles in First figure is 1 and in second figure is 4 and in third figure is 10.

Calculation:

The sequence is $1,4,10,19,31,46$

The recursive formula is $a_n=a_{n-1}+3(n-1)$

Thus, the recursive formula is $a_n=a_{n-1}+3(n-1)$

(c)

To determine

Find the number of triangles in the tenth figure.

Answer to Problem 46PPS

The number of triangles in the tenth figure is 136

Explanation of Solution

Given:

  

How many triangles will be in the tenth figure?

Concept Used:

The number of Triangles in First figure is 1 and in second figure is 4 and in third figure is 10.

Calculation:

The recursive formula is $a_n=a_{n-1}+3(n-1)$

  $\begin{array}{l}{a}_7=a_6+3(n-1)=46+3(7-1)=64\\ a_8=a_7+3(n-1)=31+3(8-1)=85\\ a_9=a_8+3(n-1)=31+3(9-1)=109\\ a_{10}=a_9+3(n-1)=31+3(10-1)=136\end{array}$

The number of triangles in the tenth figure is 136

Thus, the number of triangles in the tenth figure is 136