I work as an insurance agent and I am trying to solve a math formula
that will yield the interest rate required for a basic investment. The
formula I have is as follows:

FV = PMT((((1+i)^n)-1)/i) * (1+i)

FV = Future Value
PMT = Regular Contribution
n = Number of periods
i = interest rate

I need to solve the aforementioned formula for i. 
Example:

A future value of 15,646 with an annual contribution of 833.1 over 15
years gives an interest rate of ~ .0276060

15,646 = 833.1((((1+i)^15)-1)/i) * (1+i) ; i =~ .0276060

If I'm reading your upon question correctly, you want to modify the
formula so that it will look something like:

i= ....( ...)..(..)

Right?

Correct. The example I put in (where i = .0276060) is what happens
when I put the formula I have given in a graphing calculator and have
it solve numerically for i. I need the general case for i, like you
commented i = ....( ...)..(..).

Thank you

Well, I found the formula in its current form overly complicated. The
negative one can be replaced with a negative term and the PMT can be
placed in the numerator as multiplication and division are worked at
the same stage, so my starting formula is:

F = (-P(1+i^n))/i * 1+i

I have to run to class now, but I'll take the formula and try to work
it out while I'm there. If I get a good answer I'll post it, if not,
hope the simplified formula helps someone out.

Not sure if F = (-P(1+i^n))/i * 1+i is equivalent, however since I am
asking I am obviously lacking with my algebra skills. Let me just note
that the 1+i term is raised to the power of n i.e. (1+i)^n.

Let me also try to write the original formual more clearly and cite
what it is from.

FV = PMT * (((1+i)^n)-1)/i) * (1+i)

Not a whole lot less cluttered given the medium for presenting it, but
it might help. Additionally the equation represents annuities due.

The page, http://www.investopedia.com/articles/03/101503.asp talks
about annuities due and has the equation in more traditional format.
The equation and its discussion is near the end of the page just
before the conclusion.

For starters it appears you are dealing with annuity-due, meaning
payments made at the beginning of a term and letting it sit after your
last payment for a period, the formulas I will be talking about deal
with annuity-immediate, payments made at the end of the period and end
balance taken right after the last payment.  To convert between the
two we will have FV of n payments = FV of n+1 payments - 1 payment
thus we take the total of your future val. + 1 payment as the future
value for the equation.
To my knowledge there isn't really a definite formula the way you are
searching for, however we do have some fun ways to figure it out to a
given decimal place, probably the best we have is the Newton-Raphson
iteration method, this is given generally as x sub(n+1) = x sub (n) -
(f(x sub(n)/f'(x sub(n) specifically dealing with interest(I'll try to
make this clean):
                     [        (1+i(sub s)^(n) - 1 - k*i(sub s)    ]
i(sub s+1) = i(sub s)[ 1 + -------------------------------------- ]
                     [     (1+i(sub s))^(n-1){1-(i(sub S)(n-1))}-1 ]
where (sub x) is the subscript to i the insurance rate, n is the
number of interest terms, and k is the future value of the annuity
divided by the payment(a little more complicated then that but this is
trying to not get into too much of the math).  Basically what we do
here is take a starting value for i, in this case the best option is
    2(k-n)
i= --------
    k(n-1)
and use that as our (i(sub 1)) complete the formula untill i(sub s+1)
is equal to i(sub s) using your example...
n=16, k=(15646 + 831.1)/831.1 =19.826 this renders our first value for
i equal to 7.652/297.39=.0257305 which we can see is already close to
the wanted interest rate.  Plugging that value for i into the formula
we get
.0257305*[1 + [(1.501524 ? 1 - .510132893)/(1.463858*.6140425 ? 1)]] = .0279209 =>
.0279209*[1 + [(1.553657 ? 1 - .553559763)/(1.511456*.5811865 ? 1)]] = .0278986

Using more exact numbers and not hastily typing them into a calculater
would allow the series to converge faster, but I'm short on time.

Aslo, if the FV is unknown but present is known and you are using
annuity-immediate the following formula is used (if it is annuity-due
then we subtact a payment and have n-1 payments.
                     [        1-(1+i(sub s)^(-n) - k*i(sub s)    ]
i(sub s+1) = i(sub s)[ 1 + ------------------------------------- ]
                     [    1-(1+i(sub s))^(-n-1){1+i(sub S)(n+1)} ]
and we use starting value of i such that
    2(n-k)
i= --------
    k(n+1)
Hope this helps you, it is hard and long but that is the only way I
learned to find interest rates accurately in my 400 level interest
theory course...

Regards,

MathIsFun

Thank you for your response mathisfun. Judging by your response it
seems like the numbers to begin to converge to the true interest rate.
I am writing a program that will use this number. I will plug this
information into the program and if I can get a reasonably close
number after perhaps 15-20 iterations of the series I would like to
close this question, give you credit and move on. I will let you know
how it goes.

I have plugged the math into my program and I am able to converge on
an interest rate within 10 iterations. I would accept mathisfun's
comment as an answer if he would like to submit it as an answer and
collect the bounty on the question.

Thanks to all who helped and gave me their time. It is very much
appreciated and inspiring.

Glad it worked out for you, that is by far the most efficient
itteration for finding the interest rate out of the 5 or so that I
know of at least.  Thanks for the approval but sadly since I am not a
google guru I am not able to actually submit an answer, so looks like
you get to keep your $10.

Happy To Help,

MathIsFun