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Question
I know intuitively that n = 3 in the following equation: (n + 1)/(2^n) = (1/2).  But, I cannot solve it algebratically.  Can you show me the steps to prove that n = 3?  

Thank you,
Jimmy

Answer
The first thing to do is cross multiply.
That is, we know 2(n+1) = 2^n.
That is the same as 2^n - 2n - 2 = 0.
Try n=1: 2^1 - 2 - 2 = 2 - 2 - 2 = -2.
Try n=2: 2^2 - 4 - 2 = 4 - 4 - 2 = -2.
Try n=3: 2^3 - 6 - 2 = 8 - 6 - 2 = 0.
Yet that method only works by chance.

Make a function f(n) = (n+1)/2^n - 1/2.
Take n=1 and see that f(1) = 2/2 - 1/2 = 1/2.
Take n=2 and see that f(2) = 3/4 - 1/2 = 1/4.
Since going from 1 to 2 decreased by 1/4, we want to go one more to decrease by 1/4 again.
Trying n=3 gives f(n) = 4/8 - 1/2 = 1/2 - 1/2 = 0.

The second method got to the root as well.  If it hadn't found it, though,
continuing to use it would have found it eventually.


For example, take x^2 = 5.
The function would be f(x) = x^2 - 5.
Trying x=1 gives f(1) = 1 - 5 = -4.
Trying x=2 gives f(2) = 4 - 5 = -1.

Since a step of one move us 3/4 of the distance closer, and what we did was move 1 whole unit, that says that we need to move 1/3 closer, and get to 7/3.
Checking it out gives f(7/3) = 49/9 - 5 = 49/9 - 45/9 = 4/9.

The next number to try is in the middle, since f(2) = -1 and f(7/3) = 4/9.
The difference between the f() values is 4/9 + 1 = 4/9 + 9/9 = 13/9.
The difference in the x values is 7/3 - 2 = 7/3 - 6/3 = 1/3.

These follow the formula x[n+1] = x[n] - (x[n] - x[n-1])/(f(x[n] - f(x[n-1]).
This generates the sequence
1.00000000   -4.000000000
2.00000000   -1.000000000
2.333333333    0.444444444
2.230769231   -0.023668639
2.235955056   -0.000504987
2.236068111    5.99066E-07
2.236067977   -1.51275E-11
2.236067977    0

Checking it out, we know that x^2 should be 5.
If I use Excel to put in the last value on the left, I get x^2 - 5 = 0.
If I use it as written, there is a slight error since the compute keeps
more decimals than it shows and I get x^2 - 5 = -2.23513E-09, which is
-0.00000000223513.  That's close enough to be 0.

There is an example where the root is not so obvious.

Algebra

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