How do I calculate 4^n=1/(2(n-n^2))? Thanks.
cross multiply to get:
2(n-n^2)4^n=1
(n-n^2)4^n=1/2
Since there is no more division and the right-hand side is less than 1, we can start searching for n where n<1, say 1/2.
Try n=1/2 and see if the equation is satisfied.
Hint: one of the laws of exponents tells us that
X^(1/2)=√(X).
Is there a more elegant solution than trial and error?
You have an excellent question!
In other words, your question was more like:
Find all solutions for n such that
4^n=1/(2(n-n^2))
I do not see an explicit solution to the equation. Perhaps someone else can find one.
An explicit solution is expressed as
n=expression where expression does not contain n.
Lack of an explicit solution, I proceed as follows:
1. first bound the solution.
We can conclude that for n outside of [-1,1], we cannot have the right hand side equal to -1.
2. Find approximate solutions by graphing or otherwise. By graphing, there are two solutions, at 0.5 and 0.8, approximately.
3. proceed to refine the solutions by iterations (a glamorous name for trial and error).
Here are the details of iteration using Newton's method:
For the case of n=0.5, it is exact and so requires no further iteration.
For the case of n=0.8, we can refine the solution by setting up Newton's iteration equation:
let f(x)=4^x-1/(2*(x-x^2))
and find the derivative
f'(x)=log(4)4^x+(1-2x)/(2(x-x^2)^2)
and finally calculate a better approximation of x as
f1(x)=x-f(x)/f'(x)
and proceed to calculate
x0=0.8,
x1=f1(x0)=0.787551458195669
x2=f1(x1)=0.786114835166431
x3=f1(x2)=0.786097645437988
x4=f1(x3)=0.786097643010236
x5=f1(x4)=0.786097643010236
which means that 0.786097643010236 is our (approximate) solution.
Hope I have answered your question.
To calculate the value of n in the equation 4^n = 1/(2(n - n^2)), we can start by simplifying the equation. Let's break it down step by step:
1. Start by multiplying both sides of the equation by 2(n - n^2) to eliminate the fraction. This gives us:
2(n - n^2) * 4^n = 1
2. Expand the left side of the equation:
2n - 2n^2 * 4^n = 1
3. We can rewrite 4^n as (2^2)^n, which gives us:
2n - 2n^2 * (2^2)^n = 1
4. Simplify further by applying the property of exponents:
2n - 2n^2 * 2^(2n) = 1
Now, we have a more manageable equation to work with.
To find the value of n, we can use numerical methods or approximation techniques. Since the equation is nonlinear, there is no simple algebraic solution. One way to solve this equation is to use numerical methods like trial and error or iterative methods.
For example, you can choose a starting value for n and substitute it into the equation. Then, calculate the left side of the equation and compare it with the right side. If they are equal, you have found the solution. If not, adjust your initial value of n and repeat the process until you find a solution.
Another approach is to use a graphing calculator or a computer program to plot the graphs of the left and right sides of the equation. By visually inspecting the graphs, you can estimate the approximate values of n at which they intersect.
Keep in mind that the accuracy of the solution depends on the method used and the range of values you consider for n.