Triangle determined by arithmetic and geometric sequences

In summary: There is not a finite number of solutions.In summary, an equilateral triangle can be solved using the law of cosines and the fact that angles in a geometric sequence must add up to 180 degrees.
  • #1
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Homework Statement


Determine the triangles where the sides are consecutive elements of a geometric sequence and the angles are consecutive elements of an arithmetic sequence.

Homework Equations



The Attempt at a Solution


I don't really know how to approach this problem, what the solution would look like. I suppose it should be one or more equations that link the geometric and the arithmetic sequence.
Maybe the law of cosine but I don't know what to do with the cosines there.
 
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  • #2
EDIT: Not all of this is correct, since I misunderstood part of the description.

I don't quite have it either. My attempt right now is this:
Draw triangle. Points A, B, C for the angles. Sides a, b, c which are opposite from the points. Lower case for the sides and upper case for the vertices.

Accordingly with your given conditions,
c/b=b/a
and
B=A+1 and C=A+2

Angle Sum requirement should mean that A+B+C=180 degrees;
A+(A+1)+(A+2)=180
A=59

Now, since A is found and you expected as described arithmetic progression,
A=59
and
B=60
and
C=61.Next appears to be Law Of Sines as useful.
sin(59)/a=sin(60)/b=sin(61)/c
and this chain of equalities can be cut down by one variable
if trying a=(1/c)b^2
then
sin(59)/((b^2)/c)=sin(60)/b=sin(61)/c-----------which can be split into TWO equations in TWO variables of b and c.I have not worked further...

(See above. Some of the work is wrong.)
 
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  • #3
Robin04 said:

Homework Statement


Determine the triangles where the sides are consecutive elements of a geometric sequence and the angles are consecutive elements of an arithmetic sequence.

Homework Equations



The Attempt at a Solution


I don't really know how to approach this problem, what the solution would look like. I suppose it should be one or more equations that link the geometric and the arithmetic sequence.
Maybe the law of cosine but I don't know what to do with the cosines there.

Yes, use the law of cosines with the one angle you know.
 
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  • #4
I found it! This can only be an equilateral triangle.

First I played with the angle as https://www.physicsforums.com/members/symbolipoint.58864/ suggested.
Let the angles be ##\alpha_1 , \alpha_2, \alpha_3##
##\alpha_1## must be given by the arithmetic sequence, and then
##\alpha_2 = \alpha_1 + d##
##\alpha_3 = \alpha_1 + 2d##

As they must add up to 180 degrees, we get that ##\alpha_2 = 60^{\circ} , and \alpha_3 = \alpha_1 + d## , which was quite suprising for me.
Then I used the law of cosines and substituted the sides with the following notation:
Sides are ##x_1, x_2, x_3##
##x_2 =x_1 \cdot q##
##x_3 = x_1 \cdot q^2##

Got the following equations:
##q^2 + q^4 -2q^3 \cos{\alpha_1} -1 = 0##
##q^4-2q^2+1 = 0##
##1+q^2-q^4-q(\cos{d} - \sqrt{3} \sin{d}) = 0##

From the second equation we get that q can be 1 and -1, substituting those into the first one we get that ##\alpha_1## is 60 degrees too, so this has to be an equilateral triangle. I don't know any method to solve the third equation for d but d=0 works fine.
 
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  • #5
Trigonometry is my favourite part of math. So thanks for the interesting question and well done for solving it.
 
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  • #6
I obviously misunderstood part of the description: Arithmetic Sequence instead of Consecutive Whole Numbers. Why did I do that? Otherwise my method was good(? maybe).
 
  • #7
Robin04 said:
I found it! This can only be an equilateral triangle.

First I played with the angle as https://www.physicsforums.com/members/symbolipoint.58864/ suggested.
Let the angles be ##\alpha_1 , \alpha_2, \alpha_3##
##\alpha_1## must be given by the arithmetic sequence, and then
##\alpha_2 = \alpha_1 + d##
##\alpha_3 = \alpha_1 + 2d##

As they must add up to 180 degrees, we get that ##\alpha_2 = 60^{\circ} , and \alpha_3 = \alpha_1 + d## , which was quite suprising for me.
Then I used the law of cosines and substituted the sides with the following notation:
Sides are ##x_1, x_2, x_3##
##x_2 =x_1 \cdot q##
##x_3 = x_1 \cdot q^2##

Got the following equations:
##q^2 + q^4 -2q^3 \cos{\alpha_1} -1 = 0##
##q^4-2q^2+1 = 0##
##1+q^2-q^4-q(\cos{d} - \sqrt{3} \sin{d}) = 0##

From the second equation we get that q can be 1 and -1, substituting those into the first one we get that ##\alpha_1## is 60 degrees too, so this has to be an equilateral triangle. I don't know any method to solve the third equation for d but d=0 works fine.

If the angles are ##\alpha_1, \alpha_1+d## and ##\alpha_1+2d##, they need to add up to ##180^{\circ},## so you need
$$3 \alpha_1 + 3d = 180 \hspace{3ex} (1)$$ Any ##\alpha_1 > 0## and ##d \geq 0## that satisfy (1) will give the angles of a valid triangle, and that means that there will be infinitely many possibilities.
 
  • #8
Ray Vickson said:
If the angles are ##\alpha_1, \alpha_1+d## and ##\alpha_1+2d##, they need to add up to ##180^{\circ},## so you need
$$3 \alpha_1 + 3d = 180 \hspace{3ex} (1)$$ Any ##\alpha_1 > 0## and ##d \geq 0## that satisfy (1) will give the angles of a valid triangle, and that means that there will be infinitely many possibilities.
But there's another condition: the sides should be three consecutive elements of a geometric sequence. That cannot be satisfied only by (1).
 

1. What is a triangle determined by arithmetic and geometric sequences?

A triangle determined by arithmetic and geometric sequences is a special type of triangle where the lengths of its sides are determined by following either an arithmetic or geometric sequence.

2. How are the sides of a triangle determined by arithmetic and geometric sequences?

The sides of a triangle determined by arithmetic sequences are determined by adding a constant number to each term in the sequence. The sides of a triangle determined by geometric sequences are determined by multiplying each term in the sequence by a constant number.

3. What are the properties of a triangle determined by arithmetic and geometric sequences?

The properties of a triangle determined by arithmetic and geometric sequences include having at least one angle equal to 90 degrees, having sides that follow a specific pattern determined by the sequence, and having a fixed ratio between the length of the sides.

4. How do you find the area of a triangle determined by arithmetic and geometric sequences?

The area of a triangle determined by arithmetic and geometric sequences can be found by using the formula A = 1/2 * base * height, where the base is the length of the longest side and the height is the perpendicular distance from the base to the opposite vertex.

5. What are some real-life applications of triangles determined by arithmetic and geometric sequences?

Triangles determined by arithmetic and geometric sequences are commonly used in architecture, engineering, and design to create symmetrical and aesthetically pleasing structures. They are also used in mathematics and physics to model and solve problems involving sequences and series.

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