SOLUTION: A sequence of diagrams is formed by drawing equilateral triangles, each side is 1 cm. They give me 3 diagrams. 1. Triangle 2. Big triangle with smaller on inside ( makes 4 tria

Question 838176: A sequence of diagrams is formed by drawing equilateral triangles, each side is 1 cm.
They give me 3 diagrams.
1. Triangle
2. Big triangle with smaller on inside ( makes 4 triangles - 9 lines)
3. Big with smaller triangles ( 18lines)
Question:
The formula for the TOTAL number of one cm lines needed to draw ALL OF THE FIRST n DIAGRAMS is an^3+bn^2+n
Find the values of a and b
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
--> -->
All you have to do to draw each row of triangles is draw the numbered triangles.
The ones without numbers get formed by the sides of the numbered triangles.
You draw 1 triangle for row/diagram number 1.
You add 2 triangles for row 2 in diagram number 2, for a total of 3 triangles.
You add 3 triangles for row 3 in diagram number 3, for a total of 6 triangles.
Each triangle drawn takes 3 line segments.
The first diagram used segments, and the third used .
The pattern above continues.
We could calculate formulas for the total number of triangles in the nth diagram, the number of segments in the nth diagram, and the total number of segments in the first diagram, but we do not need to.
The problem GIVES us the formula (sort of).
If it is true,
in the first diagram, there are
segments.
We know that number is , so
--> --> .
In the first diagrams, there should be a total of
segments.
We know that there are segments, so
--> --> -->
Now we have a system of two linear equations in and that is really easy to solve:
--> .
So the total number of segments in the first diagrams is

Just for curiosity, let's se if it works for .
It would be

Since we had , , and segments in the first 3 diagrams, and , the formula works for .

Could we have deduced that formula?
Yes, laboriously, but what was required was much simpler and easier.
To arrive at that formula, we could start by calculating the number of segments in the nth diagram
You add triangles for row in diagram number .
The total number of triangles drawn to make diagram number , and that means
1-cm line segments (sides of those triangles).
is a sum of polynomials with degree 2,
so it must be a polynomial of degree 3.
We write .
From we would find out , , , and .
RELATED QUESTIONS
The vertices of triangle ABC are A(11,3), B(2k, k) and C(-1,-11). (a) Find the two... (answered by greenestamps,ikleyn)

An equilateral triangle has a perimeter of 12 cm. By joining the midpoints of its sides... (answered by ikleyn)

The fomula for area of triangle is the following: A=1/2ab Explain with drawings or... (answered by Earlsdon)

An equilateral triangle 8cm on each side is divided into smaller equilateral triangle... (answered by FrankM)

Equilateral triangle ABC has point E on altitude AD, 1/3 of the distance from D to A. If... (answered by ikleyn)

7. Which angle or angles are complementary to AOF ? (1 point) COE and FOB AOC... (answered by richard1234)

Hello! I am very confused on a question for homework, and don't know where to start. It... (answered by ikleyn)

1. Find the area of an equilateral triangle if each side measures 12 cm. A.24 cm^2 B.36 (answered by KMST)

The area of a square is a 1/4 as large as the area of a triangle. The triangle has a 16... (answered by checkley77)