Continuous Ratio Conditions: Product of 1296, Last Term 1/6 of Sum of Means

In summary, the conditions of a continuous ratio are that the product of the four terms is 1296 and the last term is 1/6 of the sum of the means.
  • #1
Joseph Richard
11
0

Homework Statement


Determine the conditions of a continuous ratio knowing that the product of the four terms is 1296 and the last term is equal to 1/6 of the sum of means.

Original question (in Portuguese):
Determinar as condições de uma proporção contínua sabendo que o produto dos quatro termos é 1296 e o último termo é igual a 1/6 da soma dos meios.

2. The attempt at a solution
x/y=y/w = 1296
w = 2y/6 , x = 1296y, and then, I tried to develop, and it gave a mess...
I would like that you guys help me.
Thank you very much in advance

Template: 18, 6, 6 and 2.
 
Last edited:
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  • #2
Joseph Richard said:

Homework Statement


Determine the conditions of a continuous ratio knowing that the product of the four terms is 1296 and the last term is equal to 1/6 of the sum of means.

Original question (in Portuguese):
Determinar as condições de uma proporção contínua sabendo que o produto das quatro termos é 1296 e o último termo é igual a 1/6 da soma dos meios.

2. The attempt at a solution
x/y=y/w = 1296
w = 2y/6 , x = 1296y, and then, I tried to develop, and it gave a mess...
I would like that you guys help me.
Thank you very much in advance

Template: 18, 6, 6 and 2.
Hello Joseph Richard, Welcome to PF !

What is the definition of a "continuous ratio"? I'm not familiar with that terminology.
 
  • #3
Hi Sammy,
I don't know if I translated wrong, but this is continuous ratio is one in which the means or the ends are the same, the party ratio and proportion
Example:
9/6 = 6/4
 
  • #4
SammyS said:
Hello Joseph Richard, Welcome to PF !

What is the definition of a "continuous ratio"? I'm not familiar with that terminology.
I would guess we have four consecutive terms in geometric series. But I'm baffled by "sum of the means". Is this the sum of the pairwise geometric means? Of the pairwise arithmetic means? Of the middle two terms? ...?
 
  • #5
Harus, I don't know too. :cry:
 
  • #6
I know, in the subject of Arithmetic, in the part of ratio and proportion, sum of the means has to do about the means and extremes.
 
  • #7
x/y=y/w
y²=xw
(y²)²=1296
y=6

1/6 of y+y ----> 1/6 de 12 => 2 = w
Replacing is
36=2x
x = 18
S = {6, 6, 18, 2}
 
  • #8
Now that I solved the question, I was on it for three days.
Thank you to everyone who helped me.
 
  • #9
Joseph Richard said:
x/y=y/w
y²=xw
(y²)²=1296
y=6

1/6 of y+y ----> 1/6 de 12 => 2 = w
Replacing is
36=2x
x = 18
S = {6, 6, 18, 2}
Should that be (18, 6, 6, 2)?
 
  • #10
Yes Haurs, I messed up.
Thank you for the correction.
 

What is a continuous ratio?

A continuous ratio is a mathematical relationship between two or more quantities where the ratio between any two adjacent terms is constant. This means that the value of the ratio remains the same even if the quantities involved change.

How do you solve for the product of 1296 and the last term being 1/6 of the sum of means?

To solve for the product of 1296 and the last term being 1/6 of the sum of means, you can use the formula for a continuous ratio: a/b = c/d = e/f = ... = k. In this case, the first term is 1296 and the last term is 1/6 of the sum of means. So we can write the equation as 1296/(1/6x) = (1/6x)/x, where x is the sum of means. By cross-multiplying and solving for x, you can find the value of the sum of means. Then, multiply this value by 1296 to find the product.

What are the implications of having a continuous ratio condition?

Having a continuous ratio condition means that the values involved are related to each other in a constant way. This can be helpful in solving mathematical problems, as it provides a predictable pattern to follow.

Can a continuous ratio condition be applied to real-life situations?

Yes, continuous ratio conditions can be applied to real-life situations. For example, it can be used to calculate the growth rate of a population or the change in value of a stock over time.

What other mathematical concepts are related to continuous ratio conditions?

Continuous ratio conditions are closely related to geometric sequences, as they both involve a constant ratio between terms. They are also related to exponential growth and decay, as the ratio between terms can be found by raising a constant value to a power.

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