solve x in equation: Using Stirling approximation, I get x=-2.825/(0.4343-log 2x)...

[lai001]

Hi. My teacher had ask us a question in class. The question is:

Given that the product of all positive odd number smaller that the number 2x is 945, find x.

I know the answer is 5, but I don't want to use trial and error method. Using Stirling's approximation, I get x = -2.825/(0.4343 - log 2x), but I am stuck there. Can anyone help me to solve this?

Thanks!

 

[Deleted member 4993]

Hi. My teacher had ask us a question in class.The question is :Given that the product of all postive odd number smaller that the number 2x is 945.Find x.I know the answer is 5,but I dont want to use trial and error method.Using Stirling approximation, I get x=-2.825/(0.4343-log 2x) and I am stucked there. Can anyone help me to solve this?Thanks.

What are the factors of 945?

 

[lai001]

I know the answer is 5,i dont want to finding factors and guessing answers, i hope to use algebraic ways like additional,multiplications,logarithms to solve for this problem.

 

[mmm4444bot]

x=-2.825/(0.4343-log 2x)

I am [stuck] there. Can anyone help me to solve this?

That equation can be written as:

log(2*x) = 0.4343 + 2.825/x

This represents a transcendental function equal to an algebraic function. There is no algebraic method to find an exact solution. You need to use technology or an iterative method, to approximate the solution.

I think it would be easier, to follow Subhotosh's suggestion. Look at the prime factorization of 945. You can then answer the question by inspection (i.e., no guessing). :cool:

 

[lai001]

That equation can be written as:

log(2*x) = 0.4343 + 2.825/x

This represents a transcendental function equal to an algebraic function. There is no algebraic method to find an exact solution. You need to use technology or an iterative method, to approximate the solution.

I think it would be easier, to follow Subhotosh's suggestion. Look at the prime factorization of 945. You can then answer the question by inspection (i.e., no guessing). :cool:

If the place that I stucked cannot be solved using algebraic method, than the entire original question also cannot be solved using algebraic method?
I personally think that inspection is just a very smart and using logical ways to guess

 

[mmm4444bot]

… inspection is just … using logical ways to guess

True, but, in this case, there's no guessing.

Maybe that's the point of the exercise, or maybe not. :cool:

 

[lai001]

True, but, in this case, there's no guessing.Maybe that's the point of the exercise, or maybe not. :cool:

Fine.....Maybe the only way to solve equations involving factorial is to use numerical methods,even using various approximation of factorial....And I actually dont know what is the point of my teacher for giving this exercise....

 

[tkhunny]

It appears to me that the question has resulted in your personal exploration and discovery. That makes it a good question.

 

[stapel]

I don't want to use trial and error method....

But sometimes guess-n-check is what is expected. Useful result: They have to have given you something with an answer that doesn't require computations into next week. The answer must be a fairly small number.

I personally think that inspection is just a very smart and using logical ways to guess

I agree! But that doesn't mean that "by inspection" is necessarily "bad".

I know the answer is 5....

Using Stirling's approximation, I get x = -2.825/(0.4343 - log 2x), but I am stuck there.

Perhaps this is making things too complicated...?

Given that the product of all positive odd number smaller that the number 2x is 945, find x.

You know that you are dealing with odd numbers, so you know that they're all of the form "2m + 1", for m = 0, 1, 2, 3, .... You know that all of the numbers are less than the (even) number 2x, so the greatest of the numbers is 2x - 1. The numbers tick off as:

m=0: 2m-1 = 1
m=1: 2m-1 = 3
m=2: 2m-1 = 5

Since the largest numbers is 2x - 1, but also 2m + 1, then the largest value of m is x - 1. Because the numbers in the product count off from 1 to 2x - 1, then the value in the middle of the product will be x.

You probably remember having seen various summation formulas for integers, such as the formula for the sum of the first so-many integers. It turns out that there is a formula for the product of the first so-many odd integers. (here). See if you can turn this formula (along with the info above) into something helpful.

And, if you're wanting a "proper proof", try using induction to prove the formula!

 

[lai001]

But sometimes guess-n-check is what is expected. Useful result: They have to have given you something with an answer that doesn't require computations into next week. The answer must be a fairly small number.


I agree! But that doesn't mean that "by inspection" is necessarily "bad".


Perhaps this is making things too complicated...?


You know that you are dealing with odd numbers, so you know that they're all of the form "2m + 1", for m = 0, 1, 2, 3, .... You know that all of the numbers are less than the (even) number 2x, so the greatest of the numbers is 2x - 1. The numbers tick off as:

m=0: 2m-1 = 1
m=1: 2m-1 = 3
m=2: 2m-1 = 5

Since the largest numbers is 2x - 1, but also 2m + 1, then the largest value of m is x - 1. Because the numbers in the product count off from 1 to 2x - 1, then the value in the middle of the product will be x.

You probably remember having seen various summation formulas for integers, such as the formula for the sum of the first so-many integers. It turns out that there is a formula for the product of the first so-many odd integers. (here). See if you can turn this formula (along with the info above) into something helpful.

And, if you're wanting a "proper proof", try using induction to prove the formula!

Sorry for late replying, but is you mean that the middle number in the above text is the median of the list of terms of the product, and 2k+1 is the number of terms?And if I use this way in my question, then I am going to solve high order equation like Quintic and above for large value?